Despise is stronger. It's stronger than 'hate' which is probably it's closest synonym Contempt suggests a less emotional feeling than 'despise' and 'hate'. Also, despise is a verb and contempt is a noun so you use them differently. We can say 'have contempt for' I despise murderers I have contempt for rich people who cheat on their tax

despise sounds a bit more serious :P

I think they're synonyms. Maybe one of them has a 'stronger feel' to it. Yeah there should be plenty, check out a thesaurus.