The main difference is speed. tr/// doesn't do interpolation or use the regex engine, so it is blazingly fast in comparison. It's very good for transliterating characters (if you've ever studied a dead language that didn't use the Arabic alphabet, you'll understand).

s/// is more flexible -- it uses the regex engine and allows character classes. It can also perform nearly abitrarily complex interpolation.

You can get by with just s///, so if you're in doubt, go for it. Just be aware that, used correctly, tr/// will run rings around it.