A difference is that most common mixers are not collision-free. For some applications of runtime hashing using ordinary hash functions, where pseudo-randomness is important regardless, being able to prove the hash function is collision-free enables useful optimizations without any prior knowledge of the keys to be hashed.

The genesis of this was database internals, which have many use cases for strongly randomized bijective function for integral types. It is used in places where most people would use an ordinary hash function.

This particular hash is not trivially invertible, since internal AES state necessary to reverse it is discarded.