Yes, they tell you different things.

A compression test is a dynamic, macro test; it checks for basic, general compression, and really only finds the big, bad stuff (broken ring, totally worn-out engine, badly leaking valve, hole in piston, etc).

A leak-down test tells you where and how much in a much finer detail. It'll find things like partially leaking rings, partially worn out cylinder wall, lightly leaking valve, etc. It also gives you SPECIFIC info, since you're putting in 100 psi (for example) and holding 95 psi.

If you wanted a quick-and-dirty check between sessions, then do a leak-down test as a baseline and then a compression test immediately afterward. From that point you can usually get away with a compression test, and when you notice a change investigate it with a leak-down to find out where it's going.

I personally don't use a compression tester, except for "holy jeebus gawd there's no compression" moments.

GA