At the same time, they are rooted in differences in methodology and principle. And even thought more than one position can be "correct", in some sense only one is "right".
One interesting group with which to illustrate the above are the Murabitun. They are sometimes called "extremist Malikis" because their shaykh (Abdalqadir as-Sufi) has some strong things to say in favor of Imam Malik and the example laid down by the early Muslim community in Medina. To be honest, I think they definitely cross the line in terms of not being tolerant enough of the other Sunni schools
From The Recovery of True Islamic Fiqh (the title almost says it all) by Abdalhaqq Bewley:
The received position regarding the madhabs is that they are virtually identical with certain insignificant peripheral differences and the whole business is really a matter of geography so that if you live in Malaysia or Indonesia you are automatically Shafi'i, if you live in India or Turkey you are Hanafi, and if you live in North or West Africa you are Maliki, and it doesn't matter which because they are basically all the same. When he investigated the matter, however, Shaykh Abdalqadir rediscovered something which proved crucial in his search for the genuine Book and Sunna. What he discovered was that the madhhabs were by no means identical and in actual fact represented quite divergent methods of deciding what constituted the Book and Sunna.
The madhhab of Imam Abu Hanifa, may Allah cover him with mercy, was formulated in Iraq, a very different environment to that of Madina al-Munawwara where the deen had been laid down, and the number of Companions who had settled there had been too few to allow a complete picture of the Sunna to emerge. For this reason Hanafi methodology involved the logical process of examining the Book and all available knowledge of the Sunna and then finding an example in them analogous to the particular case under review so that Allah's deen could be properly applied in the new situation. It thus entails the use of reason in the examination of the Book and Sunna so as to extrapolate the judgements necessary for the implementation of Islam in a new environment. It represents in essence, therefore, within the strict compass of rigorous legal and inductive precepts, the adaptation of the living and powerful deen to a new situation in order to enable it take root and flourish in fresh soil. This made it an ideal legal tool for the central governance of widely varied populations which is why we find it in Turkey as the legacy of the Uthmaniyya Khilafa and in the sub-continent where it is inherited from the Moghul empire.
[...] With Imam Shafi'i... the practise of Islam ceased to be a matter of oral transmission and behavioural imitation and became, instead, based on written texts from which the actions of the deen were derived. Imam Shafi'i's system was brilliantly devised and the Muslims owe a great debt of gratitude to him because there is no doubt that it is the rigour of his methodology which preserved so many of the sources of Islam in such a remarkable way over all these centuries.
Shaykh Abdalqadir's desire, however, was to have direct access to the Book and Sunna in their primal form as they were first implemented by the Prophet, may Allah bless him and grant him peace, and his Companions, may Allah be pleased with them, and both these methods presented the phenomenon at one remove so they were clearly not what he was seeking. It was with Imam Malik, may Allah have mercy on him, that the shaykh found what he had been looking for.
I would want to say a couple of things. First I would note that Abdalqadir seems to believe that through Imam Malik one can have "direct access to the Book and Sunna in their primal form" as if this was impossible with the other madhabs. In fact, there are other passages in his writings where he is more negative about the other schools of thought.
But secondly, I was really attracted to his description of the Hanafi methodology (at least in the above passage). I actually like the idea of Islam not being tied to a particular community in a particular time and place. It is a living and powerful deen which can be adapted to new cultural soil. There are some specific ways that the Hanafi school does give a little more weight to reason and is flexible in certain areas (while, of course, being stricter in others) In fact, I'm tempted to say that I'm a Hanafi almost in the same way that the Murabitun say they are Maliki... although hopefully I'm a lot mellower about it than they are.
For a more balanced, but still meaty, description of the development of different schools of fiqh and the differences between them, I would recommend: Source Methodology In Islamic Jurisprudence: Methodology for Research and Knowledge by Taha Jabir Al 'Alwani which is available free online.
Another good book (which I've linked to before) is the Ethics of Disagreement in Islam, also by Al 'Alwani. There are useful comparisons between the schools throughout the book, but the most concise and inclusive descriptions are found in chapter 6 on juristic perspectives.
Also here is a brief article: Which of the four orthodox madhabs has the most developed fiqh for Muslims living as minorities? by Nuh Ha Mim Keller (his answer, basically Maliki and Hanafi) Interestingly enough, apparently when Sheikh Keller had to decide which madhab to follow, the story goes, that he put the four names into a hat and randomly picked "Shafi".
And finally another good text is: The Fundamental Principles of Imam Malik's Fiqh by Muhammad Abu Zahrah. Obviously it is written from a Maliki perspective but what is interesting (to me at least) is that it gives a sizeable list of the various principles and considerations a mufti would keep in mind when evaluating whether a particular action is halal/ haram/ sunnah/ makruh/ etc... I think it gives a really good sense that fiqh is not just a matter of finding the right ayat of the Quran or finding a single hadith and acting on it. A lot of thought goes into such decisions which is why the schools developed in the first place.
And for some early comments related to the Hanafi school, check out: people of direction