When to use 2/2 or 8/8 instead of 4/4?

I never understood why someone would compose a piece in cut time instead of common time. Can someone explain it to me?

Time signatures:

Ever see those little numbers at the start of a song like 4 4 or maybe you saw a 3 4 or even a 6 8? Those are called “time signatures”. These things actually serve a very important purpose. They designate how many beats each bar will get, and also what note are we going to count as one beat. The top number tells us how many beats there are each bar. So if we’re in 4/4 we’re going to count 1,2,3,4, 1,2,3,4 and so on. What if it’s 3 /4? Well, it’s pretty simple, we’re just going to count, 1,2,3, 1,2,3 etc. Pretty easy hey? Yup! Now, the bottom number is going to tell you which note up there is going to count for 1 beat. Usually there is a 4 in the bottom, so that means that a quarter note is going to be 1 beat. What if there’s an 8 in the bottom? Well, this can get confusing, but all it means is that an eighth note is one beat.

The only difference is the unit you use to "feel" the beat. 2/2 = alla breve means you count in half notes (so it's usually a faster beat), while 8/8 supposedly has 8 distinct beats (quavers).