My answer to a question on Quora: Why do many physicists disparage mathematical rigour?
Perhaps “disparage” is a wrong word, as it has already been pointed out in this discussion. However, the question is well pointed. My answer will be short and excessively general:
In mathematics, “rigour” is a tool for, and source of, consistency of mathematical results: they remain the same no matter how they are proven, results of calculation do not depend on the way the calculation was carried out, etc.
In physics, the nature itself, the experiment is a source of consistency. The world around us is remarkably consistent.
Perhaps the same can be said with smaller words: mathematicians build the ideal world, and rigour is a way to ensure that it does not collapse. Physicists have a world ready for them, and so far it does not show any sign of collapse.