Which group of scholars differentiate between fard and wajib? What is the difference and when was it mentioned in the books? Do they represent different levels of obligation somehow?
My source is a student of knowledge at Multaqa Ahl al-Hadeeth.
Hanafi law was infiltrated by Mu'tazilite influences. The early major Usoolis of the Hanafis were the Mu'tazilites. This remained until today. It is a huge task to sieve out pure Hanafi law "pre-Mu`tazilites" in both Usool 'l-Fiqh (Fundamentals) and Furoo (branches) to determine what can be correctly attributed to the Imams of Hanafi law and what cannot.
As for Fard and Wajib according to the Hanafis, then realise that the difference as it stands at the moment is that:
Fard - an obligation established via evidence that is definite in its transmission AND meaning (i.e. Mutawaatir/Mashhur and Nass Sareeh)
Wajib - an obligation established via evidence that is indefinite in its transmission AND/OR meaning (i.e. Khabr Wahid and/or Nass Muhtamal)
Early Imams of Hanafi fiqh did not make this distinction. The later Hanafi scholars (whose creed had Mu'tazilite influences) noticed that a pattern was emerging in their fatwas, which they labelled as the "Fard v. Wujoob" issue. This was based on the "transmission of the evidence". There are little more technicalities involved which requires basic understanding of Usool 'l-Fiqh. If you are interested, you can read it here.
Apart from that, Fard and Wajib is the same thing according to majority of the scholars (with the exception of Hanafi Madh-hab ofcourse).
Fardh (obligatory), and wajib (necessary).
Shaykh Muhammad Ali al-Hanooti summarizes:
The Hanafi school differentiates between fard and wajib. Fard for them means the obligation is certain and based on clear-cut evidence. Usually it is understood by Quran or Hadith mutawatir or similar to the mutawatir.
Shah Abdul Hannan explains in his book Usul al-Fiqh:
Al-Hukm al- taklifi (defining law) may be in the form of Fard, Wajib, Mandub, Mubah, Makruh and Haram. According to majority, Fard and wazib are synonymous. If there is binding demand from the lawgiver to do something, it is wazib. However, the Hanafi's consider the demand Fard when both text and the meaning are definitive (qati) and wazib when either the text or meaning is speculative (Zanni - because liable to interpretation of meaning or investigation of authenticity). Difference between Fard and Wazib has important consequence. Denial of binding nature of a command established by definitive proof (Fard by Qati evidence) leads to unbelief. However, denial of Wazib (according to Hanafi's) or 2nd category of Fard (according to the majority) lead to transgression (Fisq).
Actually, there is one and only one classification of ibadats, and that is fard.
Source of Islam is Quran. If something is ordered in Quran, then it is a fard, it is an ibadat. Something ordered in Quran is obligatory, and must be done.
Then, what happens if something is not ordered in Quran, but Rasul orders it?
مَّنْ يُطِعِ الرَّسُولَ فَقَدْ أَطَاعَ اللّهَ وَمَن تَوَلَّى فَمَا أَرْسَلْنَاكَ عَلَيْهِمْ حَفِيظًا
He who obeys the Messenger has obeyed Allah ; but those who turn away - We have not sent you over them as a guardian.
Nisa 80 (4/80)
If Rasul orders something (without Allah ordering it explicitly in Quran) it is again fard. An obligatory to to do.
Then, what happen if something is not erdered in Quran, not ordered by Rasul, but Rasul is doing it regularly?
لَقَدْ كَانَ لَكُمْ فِي رَسُولِ اللَّهِ أُسْوَةٌ حَسَنَةٌ لِّمَن كَانَ يَرْجُو اللَّهَ وَالْيَوْمَ الْآخِرَ وَذَكَرَ اللَّهَ كَثِيرًا
There has certainly been for you in the Messenger of Allah an excellent pattern for anyone whose hope is in Allah and the Last Day and [who] remembers Allah often.
Azhab 21 (33/21)
The rightful translation of the bold faced part is "good example for those who wants to reach Allah and Day of Kiyamat".
So, doing what rasul does (but not explicitly orders) must be taken as a good example if we want to reach Allah and the Day of Kiyamat.
From the entire Quran, we deduce that, wanting to reach Allah is very important for a Mumin. Shortcut of this is to live as Rasul lived. At this point, it is important to understand what "living as Rasul lived" means. It doesn't mean doing the exact things he did at his time. One should understand his reasons in his doings, and should adapt them to his life in a perfect way. For example, Rasul Muhammad (pbuh) was brushing his teeth with a piece of wooden stick. That doesn't mean that we must prefer wooden stick over teeth-brush. There wasn't teeth-brush in his time. One who takes him as example should understand that, the reason of him using that wooden stick was to keep his mouth clean, and adapt this reason to his own live in a halal way with the best technology he can find according to his condintions (economy, social rules, etc).
So, there is only one category of ibadats; that is fard. Wajib means "required to do" in Arabic. But it is not something to categorize the ibadats in Islam. Madhabs made these kind of terms to categorize ibadats. They categorized them with terms like fard, wajib, sunnat, mustahab, mubah, etc. There is no such categorization in Quran.