This is a question that I get asked more than any other question as an Osteopath. At least one patient a day will ask this question.
The answer is not a simple one as there are many different types of Osteopaths and Chiropractors. You can find Osteopaths that treat in a chirporactic way and also Chiropractors that treat in an Osteopathic way.
The main difference is the philosophy of approach. Chiropractors generally believe that all ailments stem from the spine and the nerve that exit a specific level of vertebra. This is why they have a stereotype to 'crack a back'. Most Chiropractic treatment will involve a manipulation of some kind on your spine.
Osteopaths can also share similar idealogies but I believe are more well rounded. An osteopath can focus on articulation of a joint and also soft/deep tissue work of muscles. I find that Osteopathy is a more well rounded approach as we also can manipulate a spine just as well as a Chiropractor, but also, be able to massage/stretch and articulate the joint.
Again, this is does not mean to say that Chiropractors neglect these things, but there focus is much more on 'the crack'.
Overall (and forgive me for being bias), I believe Osteopathy is a more holistic approach to ones health. We typically spend a lot more time with patients and are able to offer the same results as chiropractors. When it comes to treatment it is personal preference to what the Patient prefers. For me it is Osteopathy all the way :).