TCS = Traction Control System
ASM = Anti-Spin Mechanism
TCS controls the traction of the driven wheels, while ASM prevents your car from spinning out/skidding
ASM doesn't give you any kind of a benefit, which is why it's recommended to turn it off. Turning it off in the options isn't enough though, as you still need to go to the car settings and put the ASM value to 0.