Mathematical Rigor--Who needs it?

Historians, philosophers and mathematicians standardly draw a distinction between rigorous and unrigorous mathematics. In drawing the distinction, they usually intend to advance some normative claim. At the very least, rigorous mathematics is supposed to be preferable to unrigorous mathematics. I want to explore the distinction between rigorous mathematics and unrigorous mathematics, in the hope of understanding why rigor is desirable, and when the need for rigor is urgent. My goal is to take my title question at face value and to answer it.