Well seeing as I'm a DO student, I think I can field this question...

Osteopathic medicine was developed by an M.D., A.T. Still. Chiropractic medicine was developed by a student of Still's, B.J. Palmer. Dr. Still, frustrated with the state of medicine in the mid-1800s, developed a number of manual manipulative techniques to treat conditions that, at the time, were not really curable with conventional (a.k.a. allopathic) medicine. To give you an example, one all osteopathic students are taught, during the 1918 influenza outbreak, patients under osteopathic care had much higher survival rates (there were neither antivirals or antibiotics for secondary bacterial infections at the time). Osteopathic physician achieved better outcomes through techniques like rib articulatory techniques (these are where the physician helps improve movement in the rib cage) and lymphatic techniques (these are where the physician, essentially manually "pumps" or "milks" the lymphatic ducts to facilitate movement in the lymphatics). None of this is rocket science really--it is well established that patients who are supine for extended periods of time (as they would have been with a nasty case of influenza) and have poor respiration are prone to things like opportunistic bacterial infections (and, death). So just by manually helping the ribs move a little better and getting improved respiration and get lymph moving again, you can (potentially) improve a patients condition (of course, we don't have any real studies on the 1918 results, it is just anecdote--but these two techniques have subsequently been demonstrated to have some efficacy).

One of the other things Still developed (or re-discovered, these techniques are pretty old) is the High Velocity, Low Amplitude thrusting techniques that chiropractors are famous (or infamous) for. This is what everyone imagines when they hear the word chiropractor--"cracking" the cervical or other spinal vertebrae. So Palmer essentially took HVLA and other manual techniques from Still et al. and ran with them, to the exclusion of other forms of medicine.

Now, let me add here, both chiropractors and, at least traditionally, osteopaths both viewed somatic dysfunctions as being a root cause of a lot of disease states. Chiropractors, or at least some fraction of them, probably obsessively focused on spinal dysfunction, whereas osteopathic physicians would take the entire body into consideration. Let me define somatic dysfunction, at least from an osteopathic standpoint. So we can start with the spine (since that is where chiropractors start). I can take any person, literally, and go up and down your spinous processes (the horizontal parts projecting from the body of your vertebrae) and find they're not aligned. No body is perfect, not a surprise. But what that means is that over the length of your spine, you're kind of rotating a little to the right in your neck, maybe a little to the left in your upper thoracics, back to the right in your lumbars. Almost everyone is twisted about the axis of their spine. Everything is connected to the spine, so that means if you're chronically (or even acutely) rotated off center, all the muscles attaching around that region of the spine are going to be put into contraction (maybe just mild). Then those muscles attach somewhere, they're going to be pulling on another group of muscles, etc. So eventually what started as a mild rotation of one vertebrae has put a number of muscles into strained positions and caused postural problems. Now the body is very robust, so this may or may not be a big deal. We kind of adapt to this and we're used to being all contorted. But you don't have to be all contorted, and, osteopathic physicians and chiropractors have techniques to "uncontort" bodies. At least some fraction of chiropractors like to address this with HVLA (e.g. the dramatic neck cracking), but osteopathic doctors (and some chiropractors) have gentler techniques (some are basically massage techniques, others more like physical therapy techniques) to get things aligned and reduce hypertonic muscles and other postural issues.

Now, there are both DO's and DC's who feel that you can treat many (most/all) illnesses through these manual manipulative techniques. The theory, at least from the osteopathic world, is that the entire body is connected (it is, literally), and local inflammation in one region (say from hypertonic muscles in spasm because of the aforementioned spinal rotation) are a drain on the bodies resources and can lead to decreased immune function or things like that. That isn't too crazy. It is physically plausible. Some DO's/DC's also think that you can effect the sympathetic and parasympathetic nervous system through spinal or other manipulations. We're getting into fuzzier ground here. And then osteopaths have things like cranial techniques, wherein we can supposedly influence the body by manipulating the cranial bones with very, very tiny movements. I will be diplomatic and say there isn't good evidence for that, nor do I see how that is possible from what we presently know about anatomy today. Which is to say there's certainly some DO's who peddle medicine that is a little on the mystical side and not so evidenced based. Of course, I feel that way about a lot of what MD's do. It is just that manipulating the fused bones of the skull sounds a lot more crazy than we're going to give a compound that we spent 10 years working on in the lab to you but we really don't know what receptors it is effecting. But it beats a placebo by 0.01%. Both magic tricks. But I digress...

So that all said, most DO's do not use manual manipulative techniques anymore. We go through more or less the same basic science and clinical training as MD students, except we get classes in manual manipulative medicine (and we may or may not get a more patient centered, wholistic approach to clinical medicine than MDs, depending on the school). We then go on to do the same residencies as MDs (we can either literally go into MD residencies, or, there is an alternate track for DOs to do DO residencies, which are more or less the same training, often in the same hospitals as MD residencies, except they might encourage you to use your osteopathic techniques). Functionally, outside of family practice, most DOs will be indistinguishable from MDs. They practice the same kind of medicine. Family practice DOs are more apt to maintain and use osteopathic manipulative techniques (especially since they can bill those). And there are also residencies for DOs in Neuromuscular Medicine/Osteopathic Manipulation, and those DOs will then go on to specialize in manual medicine (so they'd functionally be similar to a chiropractor, except they can prescribe medications if need be, and have a better background in clinical medicine). There are some studies that show that DOs are, on average, better listeners and have better bedside manner. Not a big difference, but a statistically significant one. Listening to patients has been drilled into us from day 1 (and, at least at my school, we've been doing patient encounters since the 1st week of school...so we get lots and lots of training in listening and interacting with patients).

I certainly think there is some definite utility to many of the osteopathic techniques (or I wouldn't be in a DO school to begin with). But, when it comes down to day to day practice, most DOs and MDs practice the same kind of medicine. Like I said, there are some institutional/cultural differences, and those are measurable over large enough samples. Between one MD and one DO, it is probably a wash, especially for specialties and sub-specialties. As for a DO vs a DC (the original point here), no offense to the chiropractors, but I'd rather see a DO who specialized in manual manipulative medicine. They've gone through hospital rotations, know all about conventional medicine and they know (or should know) when you should be sent to a hospital or a specialist (as opposed to trying to cure your cancer by popping your back). But I'm sure some MD will jump in and say they get better training than us (although, since we do the same residencies and often have the same lecturers--there's a couple big state schools where the MD and DO classes literally take all the same classes together--except for the osteopathic manipulation classes).

So to answer the question: by and large you can expect a routine office visit with a DO except for an FP or maybe an internist who might be able to do some manual techniques on you (if you want and it is indicated for your condition). Yes, there are some shamany-witch doctor DOs out there. There are some weirdo, woo woo MDs too (anyone from the Andrew Weil lineage, for example--I do believe he actually started a couple residencies now in complementary medicine or something like that). But those folks will often advertise as being non-mainstream or whatnot.

Lastly, you all probably want to know are DOs more or less likely to be open to your crazy Paleo ways? Out of a class of 200+ students, there are a handful of Paleo eating, Crossfitin', Vibram wearing freaks. Maybe 2-3% of the class. So the answer, again, is that your DO is probably going to be just like your MD (which is probably why there is a movement afoot to just combine the 2 and have one degree). Now, back to studying...