Is there a difference in Go between a counter using atomic operations and one using a mutex?

There is no difference in behavior. There is a difference in performance.

Mutexes are slow, due to the setup and teardown, and due to the fact that they block other goroutines for the duration of the lock.

Atomic operations are fast because they use an atomic CPU instruction (when possible), rather than relying on external locks to.

Therefore, whenever it is feasible, atomic operations should be preferred.