Qudsi hadith are those hadith that the Messenger of Allah صلى الله عليه وسلم related from Allah سبحانه و تعالى. Mullah Ali al-Qaari رحمه الله has given the following excellent definition for hadith qudsi. He wrote, "It is that which is related by the foremost of relaters and the most reliable of authorities, the best of blessings and salutations be upon him, from Allah, may He be glorified, sometimes through the medium of Gabriel, upon whom be blessings and peace, and sometimes by revelation, inspiration and dreams. Allah having entrusted to him the expressing of it in such words as he wished. It differs from the Quran in that the revelation of the latter was only through the medium of the Upright Soul [Gabriel] and is restricted to the wording specifically revealed from the Preserved Tablet." 
Such hadith usually take one of the following forms:
(1) Some begin with, "The Prophet صلى الله عليه وسلم says from among the sayings he related from his Lord..."
(2) Sometimes they begin with, "Allah the Almighty has said, from among the sayings related from Him by the Messenger of Allah صلى الله عليه وسلم ... "
(3) Or, "The Messenger of Allah صلى الله عليه وسلم said that Allah سبحانه و تعالى has said..."
(4) The words of Allah سبحانه و تعالى may be referred to as written instead of spoken.
(5) Sometimes the words from Allah سبحانه و تعالى are between or introduced by a statement of the Messenger of Allah صلى الله عليه وسلم.
(6) And, finally, an indirect but clear reference may be made to a statement of Allah سبحانه و تعالى in the words of the Messenger of Allah صلى الله عليه وسلم.
The Customarily Given Differences Between Qudsi Hadeeth and the Qur'aan
One might ask what the differences between a qudsi hadith and the Quran are. Al-Qattaan رحمه الله has noted what is commonly considered the five differences between the two. They are:
(1) The Quran was revealed by Allah سبحانه و تعالى in both meaning and wording through the angel Gabriel عليه السلام to the Prophet Muhammad صلى الله عليه وسلم. It was not received through any form of inspiration, such as a dream, vision and so forth. Hadith qudsi, on the other hand, may be received through any of those means.
(2) The Quran was established as a miracle and a challenge to all of mankind until the Day of Judgment. Hadith qudsi are not considered miraculous as such and are not a challenge to mankind.
(3) The Quran is only stated or related with reference to Allah سبحانه و تعالى. One can only say, for example, "Allah says in the Quran..." Hadith qudsi, on the other hand, are reports narrated by the Messenger of Allah صلى الله عليه وسلم and, hence, are related from him from Allah سبحانه و تعالى.
(4) All of the Quran is mutawaatir and qati 'ee ath-thaboot.  The majority of the qudsi hadith do not meet the standards of mutawaatir. They are, therefore, subject to the rules and grading of hadith. Some such hadith are sahih, some are hasan, some are weak and others are fabricated.
(5) With respect to the Quran, both the wording and the meaning were revealed to the Prophet صلى الله عليه وسلم from Allah. With respect to hadith qudsi, only the meaning was revealed to the Prophet صلى الله عليه وسلم while the wording came from the Prophet صلى الله عليه وسلم himself. Therefore, it is allowed to narrate qudsi hadith according to their meanings only (for those who allow narration of hadith according to their meanings) but this is not allowed for the Quran.
(6) Finally, only the Quran is allowed for the recitation in the prayer. Furthermore, the one who recites it will receive ten rewards for each letter of the Quran that he recites. Hadith qudsi may not be recited as part of the prayer nor is there any special reward for reciting hadith qudsi. 
Critique of the Customary Approach
There does not seem to be any room for dispute concerning (1), (2) (3) and (6) above. However, (4) and (5) need some clarification or comment. It is not correct to say that one of the differences between qudsi hadith and the Quran is that all of the Quran is mutawaatir while qudsi hadith may be sahih, hasan, weak and so forth. Many scholars state that the conditions for a narration to be considered part of the Quran are three:
(a) it must be mutawaatir,
(b) it must be consistent with at least one of the copies of the Quran that Uthman رضي الله عنه distributed throughout the Muslims lands, and
(c) it must be consistent with the Arabic language.
However, if (a) is met, the other two conditions become redundant or superfluous. By definition, (a) implies that the report is such that it cannot be questioned or doubted. Hence, what is the need for the other two conditions? Some specialists in Quranic recitation state that the first condition is not that of being mutawaatir but only that the chain must be sahih. 
Indeed, an-Namlah states what the scholars mean when they that the recital be mutawaatir is simply that it has an authentic chain back to the Prophet (may Allah exalt his mention and grant him peace),  and not the usage of the term mutawaatir according to the scholars of hadith. In this way, the other two conditions become meaningful because they ensure that, although the chain might be sahih, that no mistake has been made in the transmission of the recitation.
Furthermore, there are recitations of the Quran that are considered rejected or shaadh ("irregular"), since they do not meet the conditions stated above for the Quran. This is no different from saying that there are qudsi hadith that are weak or rejected. Hence, (3) above cannot be considered a true difference between the Quran and qudsi hadith.
Finally, if one is talking about the essential difference between the Quran and qudsi hadith, it is irrelevant to discuss whether one is mutawaatir and the other is not. Obviously, the essential differences between them existed at the time of the Prophet صلى الله عليه وسلم when the Prophet صلى الله عليه وسلم himself narrated these statements. At that time, there was no concept of mutawaatir and the Prophet صلى الله عليه وسلم would sometimes send individual teachers to specific tribes to teach the Quran. Hence, the question of mutawaatir is completely irrelevant to the issue of the difference in nature between qudsi hadith and the Quran.
There is also a difference of opinion concerning the wording of qudsi hadith. Does that wording come from Allah سبحانه و تعالى or does it come from the Prophet صلى الله عليه وسلم? As stated above in (4), al-Qattaan is of the view that its wording is from the Prophet صلى الله عليه وسلم. As-Sabbaagh is also of this opinion. He states that if the wording and meaning were both from Allah سبحانه و تعالى, then, from the shareeah point of view, it should be given the same status as the Quran. He claims that there would then be no reason to distinguish between the two wordings from Allah سبحانه و تعالى. He then argues that if that were the case, it would not be allowed to narrate a qudsi hadith according to its meaning (which the scholars say is acceptable). He also states that then it could be used in worship like the Quran, but no scholar says that.
As-Sabbaagh then poses the following question, "If the wording is not from Allah سبحانه و تعالى, why is the quote attributed to Allah?" As-Sabbaagh states that this is not problematic and is common in the Arabic language. It implies that what the quotation contains of meaning is from Allah سبحانه و تعالى and not that the exact words are from Allah سبحانه و تعالى. He argues that this is similar to Allah سبحانه و تعالى quoting the previous prophets in the Quran and saying, for example, "Moses said," while, in fact, Moses عليه السلام did not speak Arabic and those are not his exact words. 
Those quotes only contain the meaning of what Moses عليه السلام said. As-Sabbaagh then quotes Mustafa az-Zarqa as saying that qudsi hadith are like giving a person a text to translate. The translated text is still ascribed to the original speaker although the wording is not his. 
But that inevitably leads to the following question: If qudsi hadith are simply inspired by Allah سبحانه و تعالى and in the Prophet's صلى الله عليه وسلم wording, what is the difference then between qudsi hadith and the remainder of the Prophet's صلى الله عليه وسلم speech which was also inspired by Allah سبحانه و تعالى? The Messenger of Allah صلى الله عليه وسلم stated, "Certainly, I have been given the Book and what is similar to it with it." 
Furthermore, there are numerous verses in the Quran to the effect of the following verses,
"And Allaah has sent down to you the Book (The Qur'aan), and Al-Hikmah (the Sunnah), and taught you that which you knew not. And Ever Great is the Grace of Allaah unto you (O Muhammad, peace be upon him)." (Sooratun Nisaa', 4:113)
"He it is Who sent among the unlettered ones a Messenger (Muhammad صلى الله عليه وسلم) from among themselves, reciting to them His Verses, purifying them, and teaching them the Book (the Qur'aan), and Al-Hikmah  (the Sunnah). And verily, they had been before in manifest error." (Sooratul Jumu'ah, 62:2)
Allah سبحانه و تعالى also says,
"And remember (O you the members of the Prophet's family عليهم الصلاة والسلام), that which is recited in your houses of the Verses of Allaah and Al-Hikmah (the Sunnah). Verily, Allah is Ever Most Courteous, Well-Acquainted with all things." (Sooratul Ahzaab, 33:34) 
Commenting on verses of this nature, Imaam ash-Shafi'ee رحمه الله stated, "I have heard from those with whom I am pleased of the people of knowledge of the Quran who said that the Hikmah is the sunnah of the Messenger of Allah صلى الله عليه وسلم. This is the most likely opinion, Allah knows best. This is because the Quran has been mentioned and is followed by mention of the Hikmah. Allah also mentioned His bounties upon His creation by their being taught the Book and Hikmah. Therefore, it is not allowed, Allah knows best, to call the Hikmah here anything other than the sunnah of the Messenger of Allah (may Allah exalt his mention and grant him peace)." 
In other words, the sunnah as a whole is considered a form of "inspiration" from Allah سبحانه و تعالى but in the wording of the Prophet (may Allah exalt his mention and grant him peace). 
What then would distinguish qudsi hadith from other hadith of the Prophet صلى الله عليه وسلم if the inspiration in both cases is from Allah سبحانه و تعالى and the wording from the Prophet صلى الله عليه وسلم? In this case, also, there is no need for the Prophet صلى الله عليه وسلم to state, "Allah said," when, in fact, the words did not come from Allah سبحانه و تعالى.
Perhaps, this is an issue that cannot be resolved based on the information available. There is no strong evidence to definitively conclude that the wording of qudsi hadith comes from either Allah سبحانه و تعالى or the Prophet صلى الله عليه وسلم. Personally, this author leans toward the view that the wording comes directly from Allah سبحانه و تعالى.
The important point is that such qudsi hadith are known not to be part of the Quran. This is known from the Prophet صلى الله عليه وسلم himself. The Prophet صلى الله عليه وسلم would distinguish between the Quran and qudsi hadith. Whenever a verse of the Quran was revealed to him, he would call one of his scribes--and he had specific scribes for the recording of the Quran, such as Abu Bakr, Umar, Uthmaan, Zaid ibn Thaabit and others رضي الله عنهم. He would then tell the scribe to place that newly revealed verse in surah such and such. 
Furthermore, the Prophet صلى الله عليه وسلم never recited qudsi hadith in the prayers nor did he read them back to the Angel Gabriel عليه السلام when going over the Quran.
The important matter, then, is that the qudsi hadith were distinguished from the Quran by the Prophet صلى الله عليه وسلم himself. This is how it is known that such statements from Allah سبحانه و تعالى do not form part of the Quran. However, this in itself does not rule out the possibility that the wording of qudsi hadith are from Allah. The Prophet صلى الله عليه وسلم narrated them as Allah's words and this is probably the strongest indication concerning this matter. This implies that they are words from Allah سبحانه و تعالى but it was made clear to the Prophet صلى الله عليه وسلم that they did not form part of the Quran. This direct inspiration of wording is what distinguishes them from the remainder of the statements of the Prophet (peace be upon him). 
 Quoted in Ezzedin Ibrahim and Denys Johnson-Davies, Forty Hadith Qudsi (Beirut: The Holy Koran Publishing House, 1 980), p. 8.
 Mutawaatir means that the transmission has been continuous since the time of the Prophet صلى الله عليه وسلم to such an extent that it is inconceivable that all of the narrators coincidentally made the same mistake or all agreed upon a forgery. If a hadith is mutawaatir, it does not need to be graded under to the rules of hadith because it is authentic a priori. Qati 'ee ath-thaboot, "definitively confirmed", implies that no Muslim has the right to reject any part of it.
 Minaa al-Qattaan, Mabaahith ft Uloom al-Quran (Beirut: Muassasat al-Risaalah, 1981 ), pp. 25-29. Virtually the same points are also made in Sultan, pp. 210-212; Muhammad Ubaidaat, Dirasaat ft Uloom al-Quran (Amman, Jordan: Daar Arnrnaar, 1 990), pp. 53-55; Ahmad Von Dentler, Ulum Al-Quran: An Introduction to the Sciences of the Quran (Leicester, United Kingdom: The Islamic Foundation, 1 994), pp. 20-2 1 .
 This is the conclusion of the experts in Quranic recitation, Makki ibn Abu Taalib and ibn al-Jaziri. [For quotes from them, see Ahmad al-Baili, Al-Ikhtilaaf Bain al-Qiraat (Beirut: Daar alJeel, 1988), p. 77; also see Khaalid al-Sabt, Qawaaid al-Tafseer: Jamaan wa Diraasah (al-Khobar, Saudi Arabia, 1 997), vol. l, pp. 84-85.] According to al-Funaisaan's doctoral thesis, such was also the opinion of ibn Taimiya, although al-Funaisaan did not give a reference for that opinion. See Saood al-Funaisaan, Ikhtilaaf al-Mufassireen: Asbaabahu wa Athaarahu (Riyadh: Daar Ishbeeliyyah, 1997), p. 85. This was also the conclusion of Hasan Itr in his thesis, Al-Ahruf as-Sabah wa Manzalat al-Qiraat Minha (Beirut: Daar al-Bashaair al-Islaamiyah, 1988), pp. 320- 321 . This conclusion does not affect the fact that the Quran is definitively confirmed. The Quran, as a whole, is mutawaatir but not every reading need be mutawaatir to be considered part of the Quran.
 Abdul Kareem al-Namlah, Itihaaf Dhawee al-Basaair bi-Sharh Raudhat al-Naadhir ft Usool al-Fiqh ala Madhhab al-Imaam Ahmad ibn Hanbal (Riyadh: Daar al-Aasimah, 1 996), vol. 2, p. 304.
 Obviously, in such a case, that fact is understood by the listener, since the listener knows that Moses عليه السلام, for example, did not preach in Arabic. However, there is no need to make that assumption with respect to quoting what has been inspired from Allah سبحانه و تعالى. Allah سبحانه و تعالى can make that revelation in Arabic.
 As-Sabbaagh, pp. 84-86. He also makes the argument that the Quran is mutawaatir while qudsi hadith are not.
 Recorded by Ahmad and Abu Dawood. According to al-Albaani, it is sahih. See al-Albaani, Saheeh al-Jaami, vol. 1, p. 516. Hence, the Prophet صلى الله عليه وسلم received two forms of revelation. One is recited in the Book or Quran. The second, his sunnah, is not recited but it is also revelation from Allah.
 Hikmah literally means, "wisdom". However, to translate it in such a general sense in this verse may not be proper, as the text above demonstrates.
 Also see Sooratul Baqara, 2:231 for another verse with clear reference to the revelation of al-Hikmah.
 Muhammad ibn Idrees ash-Shafi'ee رحمه الله, ar-Risaalah (Ahmad Shakir, ed., no other publication information given), p. 78.
 An excellent discussion demonstrating that the sunnah is a revelation from Allah سبحانه و تعالى may be found in al-Husain Shawaat (al-Houcine Chouat), Hujjiyat as-Sunnah (Falls Church, VA: American Open University, n.d.), pp. 29-37.
 This was reported by ibn Abbaas and Uthmaan رضي الله عنهم. Such reports have been recorded by at-Tirmidhi, Ahmad, ibn Hibbaan, al-Haakim and others. Cf., Khaalid al-Ik, Tareekh Tautheeq Nass al-Quran al-Kareem (Beirut: Daar al-Fikr, 1 986), pp. 30-3 1 ; Muhammad Abu Shahbah, al-Madkhal li-Diraasat al-Quran al-Kareem (Cairo: Maktabah al-Sunnah, 1992), pp. 241 -243.
 Commentary on the 40 Hadeeth of Imaam an-Nawawi, Vol. 2, pp. 745-750