They have different criteria as to the existance of the integral. All functions which are Legesque integrable are Riemann integrable but not vice versa.

Riemann is the most common. If the function is at least peicewise continuous, the definite integral exists. If you want the indefinite integral of an otherwise integable function you have to consider the end points. If one or the other approches infinity too rapidly you are in trouble.

If the function is not continuous (goes to infinity somewhere) it may still be integrable by Riemann techniques. There are various liniting procescess whihc can be used to check for integrableness. See textbooks.