Mathematicians like to develop new math right at the boundary of what is known. Physicists don’t have that luxury because they have to describe/model/build/etc things that correlate to what is actually going on in the world.

The mathematician of this joke would scan the edge of the light, finding nothing. Then he would keep lighting little lanterns at the perimeter to make the lighted area larger until finally his keys were within sight.

The physicist in this joke would presumably root around in the dark where she thinks her keys actually were. Upon finding them through brute force and luck, she might think “wow maybe one day this place will be illuminated so I can tell wtf I just did”

Non-software-developer humans often use things called "lamps" to illuminate spaces at night. Unfortunately, illumination inhibits effective nighttime coding.

The idea is that often a breakthrough in mathematics isn't achieved by tackling a problem directly, but converting it to a simpler problem, then solving that one.

Disclosure: I'm a software developer.