What are the differences between Hanafi, Shafi, Hanbali and Maliki in Islam?

The Hanafi school of thought - was the earliest of the 4 mentioned, attributed to a student of a sahabah/companion of the Prophet s.a.a.s. However, it was founded in Iraq (intellectual capital of Islamic world. Renowned for its enthusiastic revival of ancient Greek logic) which was criticised by those living in Madinah (city of the Prophet Muhammad s.a.a.s) as having slightly different practises than the more learned ones in Madinah (those in Madinah claimed to follow traditions of the prophet s.a.a.s practised by thousands narrating from thousands (mutawattir) - rather than following a single/aHad hadith narrated from one person attributed to the prophet s.a.a.s ).

The Maliki school of thought - is the second earliest of the 4. this was the formalised the practises and interpretations of the learned ones living in Madinah (city of prophet Muhammad s.a.a.s). It draws it sources from widely accepted and practised sunnah of Madinah as a whole, which were shaped by the Prophet s.a.a.s and the sahabah whom lived amongst him (including the first caliphs like Omar r.a. and Ali r.a, etc).

The Shafi school of thought - this school comes next, and attempts to resolve issues regarding small differences in Islamic practises. So the imam collected all the hadith and attempted to categorise them into authentic, strong, weak, etc. Discarding all the weak hadith and keeping the rest: this made up the foundations of this school. (however it was criticised by the other 2 since discarding weak hadith and ignoring sayings of the sahabah - loses valuable information about the details of certain practises).

Non-the-less, this school prompted later scholars, like Bukhari, and some of his students - to do the same thing. Collect as many hadith as possible. then categorise them as authentic, strong, weak, fabricated. However, Bukhari was renowned for his insight and memory and succeeded in collecting a far wider collection of hadith. Some even say that his saheeh collection was taken as his own school of thought.

Hanbali school of thought - this school was by a contemporary of Bukhari, imam Ahmad. He was well renowned for his knowledge of hadith. his school of thought was founded on both authentic, strong and weak hadith (in contrast to some others, which did not accept weak hadith).

However, he was criticised by other well known scholars, like imam al-Tabari (renowned for works like Tabari’s history and Tabari’s Tafsir - more renowned than the infamous tafsir ibn kathir).  Tabari stated that Ahmad ibn Hanbal was excellent in his field as a traditionalist (collector of hadith) -yet not very good at being a judge (fiqh) since many held the view that the religion was not as simple as following the Quran and Hadith, but required intricate knowledge and wisdom to know how to apply the two and deal with contradictory hadith, etc.  He went on to found his own Jariri school of thought that, like so many others, eventually became extinct (one can assume though - based on his works mentioned above - that it drew upon a more wide range of sources being less concerned about authenticity - hence drawing criticism from Hanbali school - apparently labelling him as an innovator!   Tabari's approach was conciliatory and moderate, seeking harmonious agreement between conflicting opinions).

Salafi - this is the most modern movement (they dont call it a school of thought however) and it is said to be based upon the authentic hadith collected by Bukhari and a few of his students as well as others - which together make up the 6 authentic books of Sunnah (bukhari, ibn majah, etc). The majority of their scholars also do not follow weak hadith (only authentic and strong) - thus holding the same criticisms as other schools who sought to do the same
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In terms of differences of practise - they are largely on small details. For example:

In prayer, when sitting and reciting duaa:

Hanafi - only stick out the forefinger when saying the shahadah (la ilaha illallah…)

Maliki - make a fist and stick out the forefinger and wave it back and forth

Shafii - stick out the forefinger only when saying the name of God (ill..allah)

Hanbali - stick out the forefinger throughout the sitting - but do not move it back and forth

(Salafi - stick out forefinger and move it back and forth - similar to maliki)
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In terms of differences in deriving Islamic rulings:

1 - PRIMARY SOURCES =  Qur'an  (Quranists stop here)
2 - SECONDARY SOURCES = Prophetic opinions+practises/Sunnah (Authentic, Strong, Weak)
3 - TERTIARY SOURCES =  Consensus/Ijmaa of the opinions+practises of the 4 Caliphs(Abu Bakr ra, Omar ra, Uthman ra, Ali ra), Companions/Sahabah, People of Madinah, (Hanbalis stop here)
4 - LOGICAL REASONING = Analogy/Qiyas (Shafiis stop here),
   Abstraction-Deduction-Induction/Ijtihad (Malikis and Hanafis stop here)
5 - PERSONAL INSIGHT =  Wisdom/Hikmah (Sufiis stop here)