It is not so much a difference in skills, but the philosophical approach that determines their use.
A physicist would primarily be interested in questions about what reality is and how we perceive it.
A mathematician would primarily be interested in how we reason about things: in abstraction and logic, in linguistic and non-linguistic mental constructs.
This difference in philosophy changes the kinds of questions one asks and the kinds of answers that are considered "satisfactory".
It also leads to differences in the aspects of the skills (be they computational, relational, spatial or some other) that one hones even when the basic skills are from the same set.
In summary, the difference is not so much in the skill sets, but in the value/onus placed by the "owner" of these skills on them. As a consequence, their use can be different.
Update: Looking through the answers, I realised that Andrew has given a somewhat similar, but more detailed answer. However, he has written it as a physicist, and I have written it as a mathematician.
An Example: Occam's Razor is commonly used in physics research and in mathematical thinking.
In physics, it would typically be used to remove things which have low impact on the phenomenon under consideration. To find out "which terms to neglect".
In mathematics, it would be used to remove hypothesis that are not used in a certain proof or in typical uses of a certain definition.
In the former case, the reality being described determines what can be "shaved off". In the latter case, the argument or calculation determines what can be excised.