I don't like hitting off mats. Mats can hide a bad swing's impact with the ball. What happens a lot of times is the club will slide along the mat into the ball. The resulting shot looks good, but gives the golfer suspect feed back. Problem is, that if the this type of swing is used on real turf, the golfer will endure many a fat shot until corrected. For you Euro folks you might still call this "sclaffing" a shot.
However, sometimes a mat is all a golfer has to work with. I am in a situation right now where where most of my practice is done while hitting off a mat. When hitting off a mat, I am usually using my irons. I want to hear the "click" of a crisp, solid hit. This tells me, I am hitting the ball first, and not the mat. If I hear anything else, like a "thud" then I know my ball contact with the club is probably a fatty. This gives credence to the old saying "hit it thin to win".