To compete with one another in acts of righteousness and getting closer to Allaah سبحانه و تعالى is not an aspect of the disease of hasad. This, in fact, is a praised action as everyone should be striving their best to do what is pleasing to Allaah سبحانه و تعالى. Allaah سبحانه و تعالى has said,
"And for this let (all) those strive who want to strive." (Soorah al-Mutaffifeen, 83:26)
There are two important keys to this type of "healthy competition".
The first key is that one wishes he has goodness and even excels others in goodness while, at the same time, he does not mind if others have the same goodness. In other words, there is no jealousy or envy involved. He is not hoping to remove what others have received nor does it make him unhappy that others have received such good. Indeed, if he is a true believer, he will be competing with others to please Allaah سبحانه و تعالى while, at the same time, he will love that his brother is also doing what is pleasing to Allaah سبحانه و تعالى.
The second key is that the competition is with respect to the Hereafter and not for the matters of this world. Competition with respect to this world is blameworthy. Allaah سبحانه و تعالى has referred to such people in the Qur'aan,
"So he (Qaaroon) went forth before his people in his pomp. Those who were desirous of the life of the world, said: "Ah, would that we had the like of what Qaaroon (Korah) has been given? Verily! He is the owner of a great fortune." But those who had been given (religious) knowledge said: "Woe to you! The Reward of Allaah (in the Hereafter) is better for those who believe and do righteous good deeds, and this none shall attain except those who are patient (in following the truth)." (Soorah al-Qasas, 28:79-80)
Related to this topic, there is a hadeeth of the Prophet صل الله عليه و على آله و سلم which may cause some confusion. It states, "There is to be no hasad except with respect to two: a person to whom Allaah has given wealth and he uses it up for the purposes of truth. And another man whom Allaah has given wisdom and he judges by it and teaches it." 
In this hadeeth, though, the meaning hasad is actually al-ghabtah الغبطة (where one wishes for what another has but does not wish that the blessing be removed from the other person). The word hasad has been used as a type of metaphor.  
 Saheeh al-Bukhaari
 Jaami' al-'Uloom wal-Hikam, vol. 2, p. 262
 Commentary on the Forty Hadeeth of Imaam an-Nawawi رحمه الله, vol. 2, pp. 1035-1036