You could really go either way with mats on Tommy. Your vet could certainly shave the mats and, if needed, do it under some sedation (like if they were really painful or Tommy wasn't thrilled with the clipper noise). If there was an underlying skin infection, then the vet would be able to see it and prescribe antibiotics.
A groomer would be more able to do a full grooming/shave, like a "lion-cut", if you wanted Tommy to have short, even fur all over and not just the mats clipped out. But if there was a skin infection or something, then you'd need to go see the vet.
Some vets have a cat groomer on staff, which might be the best of both worlds!