Is there a distinction between finely chop and mince, as with garlic?

^ what you ^

I think minced is what you pay for in a restaurant, and fine chopped is get for free at home.

Minced is just a technical culinary term.

A fine chop is just that - chopped fine. Minced would be taking your knife edge and smashing the chopped pieces to release the liquid essence.

Not practically. Maybe finely chop is technically a bit larger than mince, but they both indicate you want very small pieces.