No, there is no real difference between these sentences. They are both correct, and they mean the same thing.
This is because of the unusual qualities of the word fun. Fun is unusual because it can behave as either an adjective or a noun, with virtually the same meaning: (something that is) amusing and enjoyable.
In the sentence "It sounds fun," fun is an adjective that means amusing and enjoyable. In the sentence “It sounds like fun,” fun is a noun meaning something that is amusing and enjoyable.
The pairs of sentences below are also identical in meaning: