Suppose it may differ from vehicle to vehicle but here is the merc line and it follows pacemaker quite closely. Appreciate that mercs are RWD
TC/ASR This is what is says on the tin. Itsa tractioncontrol or anti skid/slip reduction. Basically it will detect when one of the wheels is spinning dissproportionately to its opposite wheel and will brake the wheel which has MOST traction. Imagine nailing it around a roundabout and lifting the inside wheel. It will detect this and brake the wheel getting drive untill the other wheel has enough traction. This is brilliant in 99% of cicumstances BUT useless if you are stuck in snow or mud. It will detect the spin and actually brake the wheel getting drive so in effect slow you down. Getting up a hill will be impossible. I actually took a VW T5 into the show field at Hit the North. It got stuck and kind peeps tried to give it a push. They started getting annoyed when they said i kept stopping. It wasnt untill i noticed the ASR button that i cottoned on and turned it off. So if you want to get out of a stuck situation then ASR/TC off. I dont turn it off unless i get in a situation where i know this is important as otherwise its a blooming good indicator of road conditions as you get a warning light on the dash.
ESP - Well this is basically Electronic Stability Programme. It doesnt usually have a switch to turn it off as it is working all the time. it works on the Steering/brakes and inertia of the vehicle. Imagine driving around a muddy bend too fast. You are turning right but the muddy conditions mean the car is actually travelling straight on. This is detected and controls throttle and all four brakes to go the way you are steering. There would be little point in turning this off to gain more traction as it probably wouldnt even be doing anything in a stuck situation.
Me i would switch the asr/tc off to get going then switch every thing back on as soon as you get going