Praise be to Allah.
Fard and wajib are synonymous
The majority of scholars of usul – apart from the Hanafis and one report from Imam Ahmad – are of the view that fard and wajib are synonymous.
Fard, or wajib, is that which the Lawgiver enjoined by way of obligation, meaning that the one who does it is deserving of reward and the one who fails to do it is deserving of punishment. It is the same whether the obligation is proven on the basis of definitive or ambiguous evidence. There is no difference between them in terms of the ruling or the reward.
Fard and wajib according to Abu Hanifah (may Allah have mercy on him)
However, the Hanafis differentiate between fard and wajib. In their view, fard is that which is proven on the basis of definitive evidence and wajib is that which is proven on the basis of ambiguous evidence.
It says in al-Lam‘ fi Usul al-Fiqh by ash-Shirazi (23):
“Wajib, fard and maktub all refer to the same thing, which is anything the omission of which is punishable.
The companions of Abu Hanifah said: Wajib is that which is proven to be obligatory on the basis of evidence that is subject to interpretation, such as Witr prayer and the udhiyah sacrifice, according to their view.
Fard is that which is proven to be obligatory on the basis of definitive evidence, such as the five daily prayers, obligatory zakah, and the like. But this is incorrect, because these words may be understood on the basis of the shar‘i meaning, the linguistic meaning, or actual usage, and there is no differentiation in these three respects between that which is proven on the basis of definitive evidence and that which is proven on the basis of evidence that is subject to interpretation.”
In Qawati‘ al-Adillah fi’l-Usul (1/131) it says:
“Fard and wajib are the same, in our view.
The companions of Abu Hanifah claimed that fard is that which is proven to be obligatory on the basis of definitive evidence, and wajib is that which is proven to be obligatory on the basis of ambiguous evidence. End quote.
In al-Ihkam fi Usul al-Ahkam by al-Amidi (1/99) it says: There is no difference between fard and wajib according to our companions (Shafi‘is). The companions of Abu Hanifah used the word fard to refer to that which is proven to be obligatory on the basis of definitive evidence, and the word wajib to refer to that which is proven to be obligatory on the basis of ambiguous evidence. … The more correct view is that which was mentioned by our companions, which is that the difference in the way of establishing the ruling, so that the ruling itself becomes definitive or otherwise, does not mean that there is a difference in the implication.”
For more information, please see al-Bahr al-Muhit fi Usul al-Fiqh by az-Zarkashi (1/240-244)
Fard and wajib are binding
The difference between the majority and Abu Hanifah with regard to this issue is one of wording, which does not lead to any serious difference, because all are agreed that both fard and wajib are binding upon the one who is accountable, and that if he fails to do them then he is exposing himself to the punishment of Allah, may He be exalted.
This is something on which all the scholars agree, and this is what one needs to know of fiqhi rulings.
And Allah knows best.