Mode S is not just about the extended squitter (position information, heading, selected level etc.) but also about selective responses.
Mode A and C transponders will send a response everytime they are interrogated, either by a radar sweep or by a TCAS interrogation from a TCAS-equipped aircraft. As the responses are all on the same frequency, in busy airspace these responses may overlap, rendering both of them unreadable.
With mode S, an "all stations" interrogarion is only done every now and then. In between those, the equipment does selective interrogations, where only the station with a specific ID is asked to respond.
So in addition to making much more information available to ATC, they also help in avoiding frequency congestion.
Back to the original question. The transponder is also switched to STBY to verify you have the correct code set (1200, 7000 or an ATC-assigned code) during the pre-flight checks. The only reason I can think of to ever use OFF is when the aircraft doesn't have a Master Avionics switch, and you want all equipment to be off during engine start and shutdown (due to the potential for voltage spikes).
And another reason to use ON instead of ALT could be that ATC has found your altitude readout to be wrong, for instance because your encoder broke. They will ask you to turn off your altitude readout, but will want to keep your mode A 4-digit code visible. That's what you achieve in the ON setting. And for that same purpose, even mode S transponders have an ON and an ALT setting.