If you have some source or information to give to explain it would be a help. How I understand it Mutazilism or Mu'tazili was an entirely different group. Which I may add later as some form of school.
http://en.wikipedia.org/wiki/Divisions_of_Islam#Mu.27tazili
Yes I do, but first I will explain.
Mutazilites were a philosophical group who used the principles of Greek rationalism in Islamic Theology. They were revolutionary by discarding of the traditional sunni and shia dogmatics and religious sources (like the sunna, which is made of hadiths and which is the basis of the sharia). They only treated them as historical sources for inspiration, but much more, of criticism. They were after all, anti sunni/shia. Some people call them, the continuation of the kharijites, and there is indeed a relation. But its like calling Chinese communism a version of Marxism, which it clearly isnt, although there are many elements which are related politically
For the mutazilites, the relation with the khariji is more political and less theological, although the mutazilites did consist of different, many opposing, sects also.
The golden era (yes u read it clearly!) the G-O-L-D-E-N era of islamic civilization was under the rule of the mutazilites, through the Abbasid Caliphs Al Ma'mun, Al Wathiq and Al Mu'tasim. Almost ALL of the great thinkers, scientists, philosophers, with very few exceptions, were mutazilites.
These have accepted the mutazilist ideology as the state religion and installed an inquisition to wipe out sunnism and shiism and spread mutazilism. (see the word of MIHNA) When the opposition of the sunnis, who could count on vast reserves of illiterate farmers, nomadic people (bedouins) and other frustrated people, the mutazilites softened their approach to the Shia, and even till 1 point, allied with them agaisnt the sunnis. After the fall of the mutazilites by caliph Al Mutawakkil, who (re-?) installed sunnism as the state religion and reversed the inquisition and started to try to wipe out mutazilites and shia's, the mutazilites managed to take power once more in the muslim world, but only temporarily and with help and alliance with the shia of the Buyid dynasty in Iraq.
After the coming of the sunni Selcuks, and especially cause of the book burning of Rey by the Selcuk shah their role in islams political (but not theological!) history was over. Today mutazilism is only known out of sunni sources of criticism against the mutazilites.
The sunnis didnt manage to kill off rationalism, so they adopted it and sects like the Ash'arites came to being who mixed traditionalist sunni dogma with mutazilite rationalism. The Ash'arite sect died out too and a more conservative sunnism under the Selcuk, Saladin and later, the Mamluks and the Ottomans, became mainstream.
Islam in Morocco and Spain have seen points of mutazilite rationalist success too, especially thanks too Averroes (Ibn Rushd), but they died out too because of sunni anti intellectual narrow mindedness and radicalism.
What causes the misunderstanding that the mutazilites were not a completely different group, is that many sunnis refer to the golden era of islamic civilization as their own, and so, calling mutazilites a "school of thought", but it was much more than that. It was an independent political organization too (with many sects of course), but with 2 mainstream schools (Basra school and Baghdad school) who had power over armies, bureaucracy, politics and even religious affairs, it was the state religion of the Abbasid Empire under 3 caliphs. They even installed an inquisition to wipe out sunnism and shiism (but that changed as I told), and they are called in ancient sunni manuscripts as heathens who's blood is legal to spill. So dont let a sunni apologist fool you, its no "different theological school", its much more than that. It was a state, an institution opposed to allmost all of sunnism and shiisms beliefs.
Today, there are some sects/schools of thought or streams, which are actually continuations of the mutazilite rationalist cause. These are called Quranists, who only accept the Quran as the legitimate source in religion and discard everything else (like protestantism). These are very rationalistic and appalling to mainly the intellectuals in the muslim world, but they are becoming increasingly popular cause of the freedom of thought and practice and discarding of dogma and ritualism which it causes to bring.
I would divide the most important islamic factions in history like this, but plz mind that the order of counting doesnt say anything about its relevance or importance in islamic history:
1. "Sunnism" (Qurayshi elite and followers)
2. "Shiism" (Alawi elites and followers)
3. "Kharijism" (Populist Uprising against political schism of qurayshi (sunni) and Alawi (shia) elitist wars)
4. "Mutazilism" (Golden Era, rationalists influenced by hellenist rationalism and some elements in kharijite political thought)
5. Ismailism (Assassins, Fatimids, radical shia against all non-ismailis, against mainstream shia too)
6. Neo-Sunnism (Hanafiya, Hanbaliya, Shafiiya and Malikiya, mainly developed under the Ottomans)
7. Neo-Shiism (12 imami shiism after the Safavids) Khomeinism is a modern result of this safavid neo-shiism.
8. Wahhabism (sunni reformists, discarding all of sunni theology except the quran
and the sunna and reinterpreting sunni islam by these sources only. Rebelled against the Ottomans under Muhammad abd al Wahhab with British Help, seen as traitors by almost all sunni muslims nowadays cause of treason against the Caliph in Istanbul, also state religion of Saoudi Arabia) Out of wahhabist theological and political thought came qutbism and jihadism and salafism, etc..)
9. Neo-Mutazilism or Quranism (muslims of all sects who discard most of all of their sectarian beliefs in favor of rationalism, by which the main point of recognition is accepting only the Quran as the legitimate source of religion which is explained in a rational, sometimes (pseudo-) scientific way. Mostly popular among intellectuals, but becoming increasingly more popular among common people cause of the appalling freedom of thought, rationalism, and easy adoption of many principles of (european) enlightenment. But dont misunderstand, they can be very radical in their political views. Many are pan-islamists, and although they feel a deep grudge against sunni and shia reactionarism, they cant wait till they will set many things straight with western (neo-) imperialism. Many pan-arabists, pan-turkists, or turkish and arab nationalists, even Iranian (nationalist) intellectuals and Pakistani, are quranists.
So....
that was my 10 cts