The Cinque Terre is more intimate, and hiking along the coast is absolutely breathtaking. Vernazza's my favorite home base of the 5.

The food's better on the Amalfi coast. Also, you can also get to Pompeii/Herculeaneum from the Amalfi coast, so if you really want to see more ancient Roman ruins, that might tip the balance in it's favor. (Some people get burnt out on ancient things after visiting Rome Florence and Venice.)

Either one is an excellent choice for your exposure to Italy that's not Rome Florence or Venice. I say pick one, and then go to the other one the next time.