Praise be to Allah.

The majority of scholars of usool – apart from the Hanafis and one report from Imam Ahmad – are of the view that fard and waajib are synonymous. 

Fard, or waajib, 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. 

However, the Hanafis differentiate between fard and waajib. In their view, fard is that which is proven on the basis of definitive evidence and waajib is that which is proven on the basis of ambiguous evidence. 

It says in al-Lam‘ fi Usool al-Fiqh by ash-Shiraazi (23): Waajib, fard and maktoob all refer to the same thing, which is anything the omission of which is punishable. 

The companions of Abu Haneefah said: Waajib 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 zakaah, 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. End quote. 

In Qawaati‘ al-Adillah fi’l-Usool (1/131) it says: Fard and waajib are the same, in our view. 

The companions of Abu Haneefah claimed that fard is that which is proven to be obligatory on the basis of definitive evidence, and waajib is that which is proven to be obligatory on the basis of ambiguous evidence. End quote. 

In al-Ihkaam fi Usool al-Ahkaam by al-Aamidi (1/99) it says: There is no difference between fard and waajib according to our companions (Shafaa‘is). The companions of Abu Haneefah used the word fard to refer to that which is proven to be obligatory on the basis of definitive evidence, and the word waajib 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. End quote. 

For more information, please see al-Bahr al-Muheet fi Usool al-Fiqh by az-Zarkashi (1/240-244) 

The difference between the majority and Abu Haneefah 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 waajib 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 are agreed, and this is what one needs to know of fiqhi rulings. 

For more information, please see the answer to question no. 127742

And Allah knows best.