I would like to know what is the difference between research in theoretical physics and pure mathematics. In particular, what does a theoretical physicist actually do all day long for his research? In other words, what does research in theoretical physics involve? Do theoretical physicists work only on big problems like string theory or any other major topics or there are other topics which are more fit for, say, PhD dissertations?
First, you can be sure that a PhD student will almost always work on a very particular and nuanced topic in either of the fields. Even if you do a PhD in string theory, you might just ask about some very specific class of solution or about a prediction of observation of a very special measurement. It is hard to find "big" questions which haven't been asked yet and attempted to answer in the exact same way as you would attempt to.
Science is just not "waiting for the next Einstein", it is team work.
The line between mathematics and theoretical physics gets blurred especially in the nowadays "beyond standard" theories. Generally, we can say the theoretical physicist wants to predict a particular result connected with observation based on theoretical constructs and the mathematician wants to find general truths about theoretical constructs (which may have nothing to do with reality). The part where these two meet is often called "mathematical physics" but it should be clear that the conventional line might often be unclear.
As to what does a theoretical physicist do all day, I actually don't know from personal experience as I do research only part time so far but my research includes mainly formulating problems, trying to tackle them analytically, running numerical simulations and drawing physical (observation-like) conclusions. However, there will for sure be theoretical physicists whose work tends to get more analytical or numerical. For more information, it would perhaps be convenient to ask a more experienced scholar in either of the fields.
I'll take a stab at each of your questions one by one:
How does the research in theoretical physics differ from mathematics?
Research in physics and research in mathematics are very different activities. While they both use mathematics as a tool to communicate ideas, they are doing so to accomplish very different goals. Remember that conclusions in physics are expected to eventually boil down to something that can be confirmed or refuted by experiment while mathematics has no such requirement. We say that physics is an experimental science and so the problems worked on by a theoretical physicist are rooted in a physical problem.
That said, if the phenomenon being studied by a physicist is easily described using heavy duty mathematics then the theoretical physicist will do so if he's got the math chops for it (and they usually do). Often it can seem like there is no real difference at all with what a mathematician is doing and many physicists will even go so far as to create the mathematical tools they need to properly frame their question. At that point they're arguably doing mathematics -- and that's ok. I've got guys like Witten or Feynman in mind here.. The physicist is still just using the tools that are appropriate. Nevertheless the research the physicist is pursuing has a completely different motivation than the research a mathematician chases down.
What does a theoretical physicist actually do all day long for his research?
That totally depends on the subject the guy specializes in.
I knew a guy in grad school who spent the better part of two years coding up stuff in Mathmatica to calculate higher-order loop corrections. He definitely thought of himself as a theoretical physicist.
Another guy I knew worked in the nuke theory group and studied relativistic heavy ion collisions. He spent his time reading papers from the experiments (PHENIX, STAR, etc.) and examining published data to support his own position on the existence of quark-gluon plasmas and explanations for the observed transverse momentum distributions. He also thought of himself as a theoretical physicist and eventually finished up his degree to take on a post-doc in what most people think of as theory.
Yet another person I knew was very much into relativity and spent all his time working on curvature calculations. This dude was like a human calculator... Not much in the way of experimental conclusions I'm afraid but lots of keeping up with Phys. Rev D papers, studying differential geometry, and so on. Another theoretical physicist in the making...
The point is that what a theoretical physicist does all day long is totally dependent on their area of physics. It's hard to make general statements that can up an answer to your question.
Do theoretical physicists work only on big problems like string theory or any other major topics or there are other topics which are more fit for, say, PhD dissertations?
Take the three guys I mentioned above:
I'd say even though these appear to be bite sized chunks of work in their respective fields they were really all tackling the big picture stuff: renormalization, quark-gluon plasma, and the field equations. So perhaps it's just a matter of perspective.
The fundamental difference is, that the physicist can develop stuff based on heuristics and physical intuition, while the mathematician has to prove every single step he does. The work of justifying the stuff the theorist does, is again the mathematician's part, the theorist is happy as long as his theory works. For a nice example for the difference, I suggest you study the history of the Dirac $\delta$-"Function". Thing is basically, that Dirac introduced it because it seemed physically intuitive and right for him, but it took the mathematicians more than 10 years to justify it.
Just an extra thing to add: take a look at Feynman's wonderful point of view on this subject. It's very similar to what Daniel and Void answered, in my opinion- just more interesting (no offense to Daniel and Void- I think Feynman is more interesting than everyone!).