Sheikh `Abd al-Rahmân b. `Abd al-`Azîz al-`Aql, professor at al-Imâm University, Riyadh
In brief, these are two approaches to determining the principles of jurisprudence and legal theory (usûl al-fiqh).
The school of the mutakallimîn seeks to derive juristic principles from primary evidence and then use these principles to determine specific legislation. This is why the word mutakallimîn is used to describe this approach, since it refers to scholasticism and the practice of deriving conclusions from general principles that are founded upon rational and textual evidence.
This is the approach used by Shâfî`î, Mâlikî, and Hanbalî legal theorists. It is also the approach of the Zâhirî theorist Ibn Hazm in his books on jurisprudence.
The school of the fuqahâ’ seeks to derive juristic principles from the body of legislation that has already been produced by earlier jurists. They work backwards from the existing corpus of law in an attempt to deduce the principles that those jurists must have been using to derive those laws. The word fuqahâ’ means “jurists”, and the school is referred to by this word since it derives its broad legal principles from the actual rulings of particular jurists.
This is the approach used by Hanafî legal theorists. The scholars whose legal rulings they focus on are primarily Abû Hanîfah, Abû Yûsuf, Muhammad al-Shaybânî, and Zafr.
And Allah knows best.
Source: Islam Today