Let's take for example Frost White on Classic and Metallic finishes. What are the differences between the two?
Generally, the Metallic paints have more of a sheen to them and are somewhat reflective, classic paints are glossy but generally aren't reflective. Classic paint is most similar to cars in real life, and is the default color type for the vehicles NPCs drive.
There's a similar discussion about this on Reddit too.
And here's a car with a classic paint applied. You can see that this is still kinda glossy, but comparably flat to metallic paint:
Even now, these images don't quite display the difference fully. The qualities of the two paints is even easier to recognize in-game. If you have the money to spare, purchase paint in the game and compare it to see for yourself how the two coats differ.