Why do planes sometimes make a touch-and-go?
To practice approaches, traffic patterns and landings without the added downtime of taxiing back for another takeoff.  Pilots have to maintain proficiency in these skills by practicing them on a regular basis and touch and gos are some of the best ways to do this.  It can also be used for meeting regulatory currency for carrying passengers depending on local time and landing gear configuration of the airplane.  In terms of gas consumption, it’s no less wasteful than a stop and go or a taxi back.  It’s not just for small aircraft either.   I’ve seen plenty of large military aircraft doing pattern work like that as well as there is no substitute for the real thing.  Airliners occasionally do touch and gos, but generally as part of a test flight after major maintenance; company pilots will fly approaches and landings in a level D full motion simulator simply because it’s cheaper and can simulate a wide range of conditions and emergency contingencies as well.