Answer: You're right. The two verbs, which mean "to laugh," mean basically the same thing. Although you will find some regional variations, reírse is the more common of the two. Thus, while reí would be understood to mean "I laughed," it would be more common to say me reí.
There are at least two cases where one form is required:
Reírse de: When followed by de, the reflexive form usually means "to make fun of" or "to laugh at":
- Me reía de mi hermano. I used to make fun of my brother.
- Se reirán de su falta de sofisticación computarizada. They will laugh at your lack of computer sophistication.
- Me hace reír. She makes me laugh.
- Austin Powers no me hizo reir más de una vez. Austin powers didn't make me laugh more than once.

