There are really two camps of physicists. The theorists, and the experimentalists. There is an emerging third group of people, mostly computational physicists, who could be considered a mixture of these two categories.
The 'proofs' used by experimental physicists, often rely on intuition, and can be very hand-wavy, or even mathematically wrong. However, the theoretical physicists are not much different from mathematicians. Many are motivated by the connection with reality, and some are slightly less rigorous, but this is really it.
Mathematics and physics are both governed by a set of axioms. We try to make these axioms as basic as is practical. For example, "the speed of light is the same in all reference frames". In physics, these axioms could be wrong, but they are usually supported by an abundance of experimental evidence.
In math, we have things like the field axioms. For example, 0*(any number) = 0. These are true in the sense that they are definitions. However, Godel showed that having a complete and consistent set of axioms for all of mathematics was impossible.
