Why does Buddhism have so many variations, like Zen, Nembutsu, True world, Nichiren Buddhism, etc.? Why is Buddhist philosophy not uniform?
Buddhism changes as it goes into different cultures and adapts and is influenced by local and preexisting culture. It changed when it went from Indian into China and again into Japan and again into Tibet. I believe it is changing still as it goes into the West. For instance my group the Triratna Community is an explicit attempt to bring Buddhism to the West in an ecumenical way.
To follow on from that, the Trirantna's founder Sangharakshita identified the act of going for refuge (to the Buddha, Dharma and the Sangha) as the central act of all Buddhist schools and the principle that can (and does for us) underpin any Buddhist movement. So our interpretation is that all schools have the same principle so are not as divergent as one might think at first glance. Of course I'm not banging the drum for that specifically here and people are obviously more than welcome to agree or disagree on that interpretation.
Buddhism also does change within a culture by reacting to the Buddhist school before it. The most notable example I guess is the growth of Mahayana as a reaction against the perceived ossification of the Hinayana (though this term is derogatory) schools. Parallels can easily be seen in other religions such as the Protestant Reformation, the Cathars, Lollards etc...
Also I might quibble with the premise of the question itself. I'm not convinced that Buddhism does have a particular high amount of branches as compared to other religion. It's just that we are not used to the religion. Also if we don't identify culturally with any particular branch we see them all. I think that Christianity has a lot of sub divisions - Protestant, Catholic, Greek Orthodox, Methodist, Church of England etc.. Also having listened to Peter Adamson's excellent podcast of the history of philosophy, I find the branches of Islam bewildering.
This is probably a bit TR;DR. Sorry
The answers above give good cover to a complex phenomena. In the West, monastic Buddhism, either Zen, Tibetan or Theravada, seems to be showing a certain Christian flavour. Lay groups seem to be showing a post "sixties" flavour of new age, psychological, neo pagan or yoga influence, no doubt after reading the Calama discourse, so called charter of free enquiry.
Practitioners will always benefit if the four noble truths are dominant in these respective offerings; dependent origination; fourfold setting up of mindfulness; a simple acquaintance with a version of the life of the Buddha; evolution of the Sangha and Vinaya, particularly the Ovada Patimokkha and Parinibbana admonitions. In this sense, all schools are pretty much the same, leading to the same goals.
To answer directly on the question why buddhist philosophy is not uniform: it was not the Buddha's intention, to teach a complete and uniform system of philosophy, quite on the contrary, the historical Buddha refused to answer questions, which he did not think conducive to liberation. I will provide two scriptural passages which underline this:
In the Cūḷa-Māluṅkya-Sutta, (MN I.63), ven. Māluṅkyāputta complains to the Buddha, that his preaching leaves out some of the deepest and most important questions, such as whether the world is eternal or confined in time, whether the tathāgata lives on after death and so on. The Buddha in turn asks him, whether he has ever claimed that he was going to answer these questions or whether Māluṅkyāputta had agreed to follow the Buddha on the condition that he would answer these. Both of them are answered in the negative. With a simile the Buddha then goes on to explain, that knowledge of these things are not conducive to the Buddha's goal, the end of suffering.
In SN V.437f the Buddha asks while in a forest, grabbing a hand full of leaves, whether the leaves in his hand are more numerous or those still up in the trees. Receiving the obvious answer, he states, that similarly what he has taught is much less than what he has chosen not to teach. That, which he has taught is just what is conducive to the aforementioned goal.
Thus the early monks were not furnished with a complete system of philosophy and moreover the Buddha, since he was not discussing philosophy in the first place, but preaching, adapted his sermons to the varying listeners. So not only the system was not complete, it also seemed to contain contradictory statements, such as ignorance as the root of suffering in one place (for example in pratītyasamutpāda) and in another thirst (like the noble truths) and suchlike.
I'm not saying that there are more differences in philosophy in Buddhism than in any other religion, but this is precisely why they are there in Buddhism. The early followers were engaged to reconcile seeming contradictions and to furnish the teaching with a sound philosophical base, such as ontology, epistemology, metaphysics. The intellectual climate in India in that time being very liberal, there were lots of different answers, that were accepted and rejected, discussed and disseminated.
Moreover, some scholars (for example Bronkhorst) insinuate, that maybe not only had the teaching to be safe in an inner-Indian competition, but that also the presence of Greek philosophy in the outskirts of India (Iran, Bactria) very much encouraged a philosophical development of the teaching. This is suggested by the fact, that the Sarvastivadin branch of Buddhism, which was situated mostly in the northwestern regions, shows the "most philosophical" development, trying to fathom the philosophical implications of such teachings as anātman.
Note, that this refers to early buddhist schools of philosophy, but accounts also in a certain way for the appearance of Mahayana Buddhism.
Historical Buddha turned the Wheel of Dharma three times. In each turnings he gave different teachings. During the first turn, he taught about the Four Noble Truths, during the second one - emptiness and the last one - ultimate Buddha Nature (some schools may categorize it slightly differently).
Overall, Buddha left a vast amount of teachings with one wish - that they will help people to reach lasting happiness. The variety of teachings (and nowadays schools) reflect the abilities of us, learners. Buddha was teaching simple farmers, philosophers and kings and he knew that all of them need different approach. Buddhist teachings can be compared to a pharmacy - there is a lot of medicines, but a pharmacist will dispense to you only the ones which are suited for your illness. Taking other medicines might be unnecessary or even harmful. At the end we want all the customers to be healthy regardless of the means they used.
This is a very good question -- I find myself pulled in two different directions in answering.
First, my reaction is "well why would we expect it to be otherwise?". All religions branch, so of course Buddhism does too. Branching is a function of the humans who interpret and pass on the teachings through the generations.
But then I think, well why doesn't science branch? Where religions seem to diverge, scientific understanding converges (with occasional bumps here and there). To the extent that Buddhism is essentially a science of reality, the expectation that it not branch is a reasonable one.
My conclusion is that Buddhism, like all other similar systems of thought/practice, branches to the extent that we overlay the underlying "science of reality" with distorting human constructs and artefacts. But to the extent we can remove those -- c.f. Aldous Huxley's "Perennial Philosophy" -- we will see convergence. Here is a superb talk proposing a view that we are nearing the end of a paradigm (a la Thomas Kuhn) in which that underlying science has indeed been overlaid with cruft. I'd even offer that the very existence of buddhism.SE is an example of that ending! :-)
The lack of uniformity within Buddhism or Buddhist traditions should not be surprising at all, when you consider the dharma teaching on impermanence. A tradition requires the practitioners to share with each other over many generations to survive. As environments change those changes affect how practitioners pass along their traditions from one generation to the next. After reading exerts from “In the Buddha’s Words: An Anthology of Discourses from the Pali Canon” by Bhikku Bodhi, it is easy to see how the various Buddhist traditions developed in reaction to individuals’ interpretations of his words based on their current circumstances.