There are several main points in which Buddhists and Christians diverge.
Christians believe that there's a world out there, a Creation, that has solid, ultimate existence. We are part of that creation. God, an omnipotent, all perfect and eternal being created everything. This bares two significant differences, one ontological and other epistemological.
In terms of ontology, things ultimately exist and are real by their own and so do we. The same goes for God. In terms of epistemology, we can understand the Creator by studying his creation. This were the first steps of science and then God ended up being excluded from the equation, but that's a different matter. The worldview is nearly the same with the exception that the later excludes the first cause, God, and replaces it by a natural event, the Big Bang. God's laws are replaced by the laws of nature, and everything that exists is said to be physical and not needing the interference, past, present or future, of any supramundane being (physicalism and the principle of closure).
Among scientists and Christians there are divergences about all this, but I believe this is the mainstream opinion.
There's also a difference soteriologically speaking. We were born in sin and need to save ourselves through Christ. The purpose is to elevate our immortal soul to heaven and live there forever. If we fail, and that can be just by believing in things we shouldn't, we are doomed to hell forever. For that matter our beliefs are very important, perhaps even more than our actions.
In Buddhism there are different ways of seeing reality, but it's always considered an illusion. We don't see things as they are. Everything is empty. So things lack true existence and are always dependent of other factors. Our own self is just that, an illusion that we take for real. This self we grasp doesn't exist as something of its own. This bares two opposite views in terms of ontology and epistemology. There's no Creation, but a set of processes that generate our experience and shared experience of what we think to be truly real (samsaric experience) and even if studying nature can be helpful, this doesn't allow the practitioner to reach Nirvana or Enlightenment. To do so he needs to follow the Noble Eight-fold Path.
Soteriologically speaking, our main problem is innate ignorance. Due to it we see a warped version of reality. This ignorance branches in a myriad of factors that lead us to act in a way that is not in accordance to our best interests. So our actions generate future consequences (this is karma) and we end up binding ourselves more and more to this corrupt experience of reality. This deluded cyclic experience is called samsara. Life and death are inside samsara. Although we can't say there's a being, a soul, changing bodies through several rebirths, one life succeeds the former as a consequence of all the previous actions. What remains is a mental continuum, not a thing independently existent. A continuum of causes and consequences. We may reborn as beings other than human. Human rebirth is considered the most favorable for the practice of Dharma. Nirvana or Enlightenment is the way out of samsara, out of delusion.
Here there are some major differences between schools and traditions.
You'll find very different versions of Buddhism. The interpretation of the meaning of emptiness and the consequences of this vary. So does the definition of Enlightenment and this too has consequences. So you can have a school considering that all we see are appearances and there's nothing real about them while others will say that there are small entities that do have existence, even if only by a very short period of time. Some will say that the end of the path is Arhatship while others will say it's perfect Buddhahood. Some will prescribe a set of methods emphacizing mainly conduct while others will put more emphazis on transcendent wisdom. As some Buddhist schools developed nearly or completely isolated for many centuries, you'll find some major differences. Still, there are things in which they all agree.
You can see a list of these points here
That's it in a nutshell, I guess. It's very hard to speak of "Christianity" and "Buddhism" as wholes, because of the differences that exist between movements encompassed by these two broad labels.