I spent two weeks at the beach

I spent two weeks at the beach
 

And I spent 2 weeks on the beach.
Difference?

On the beach can be nautical slang for not at sea. In the sense you probably intend, there doesn't seem to be much difference; although personally I favour at, the small number of examples in BNC favour on.

In the U.S., on the beach literally means on the sand. At the beach means staying by the seashore.

Examples: The shipwreck survivors were found stranded on the beach.

I just spent a week at the Beach. It was hot in Miami.