This is just kind of a difficult question because the distinctions aren't always as clear as your question implies. For example, you don't see people walking around the company or the University wearing a tag that says "Mathematician" or "Physicist" underneath their name. Many times the mathematicians work on physical problems (e.g., "applied math") and yet you will find as well, physicists working on mathematical results.
For the sake of discussion, let's just assume it's all black and white and clearly demarcated between "math work" and "physics work". (To reemphasize: this is not at all accurate.) Then the partitioning of work between these people really all depends on the project, people's personal background, and so on and so forth. There is one key difference in that the mathematician is typically expected to have less intuition for the physics of the problem. Thus the mathematician will typically work on more methodological aspects (e.g., programming, numerical stability, etc.) which do not require a physical interpretation. The physics person will be more responsible for analysis of the data results and interpretation of the "conceptual meaning" behind whatever is going on.
