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Assuming that I am capable of doing either, what are the pros and cons of either route?

I know that mathematics is essential to doing physics, and in most math courses, there are applied math modules that are very much related to physics. Also that many research physicists have math degrees. But surely there is a reason why people choose the physics course over the math course and vice versa? In particular, what are the advantages of a physics degree over a math one, given that a math degree could lead to PhD in physics and possibly open more possibilities in say the financial world (though that is not my interest)?

How important is experimental experience to a theorist?

Thank you.