The most obvious difference is the continuous Pmax power rating for heat dissipation.
5 watts MOV vs 0.25 watts TVS
Power vs signal vs location affects the disturbances and solutions available. This will not be a tutorial.
Peak current MOV is better for high AC line surges for locations with unbalanced lines so 100A half wave at 60Hz
Consider where one phase might have a fault or short resulting in the other phase possible have a half cycle overvoltage from a delta-Y distribution or a 3 phase transformer. , this then ought to blow a designed-in fuse. This is important in areas of high fault occurrences.
TVS is better for lightning surges 20us impulses so 1500A. But not 10kA. Peak voltages may be reduced with a line filter or a CM choke or CLC PI filter and then TVS V threshold can be lower towards some typical 20% over application voltage.
TVS may not be suitable for AC grid due to overvoltage durations of a half cycle or more but excellent for narrow transients like 20us or signal lines due to lower differential resistance and faster response time.
Best followup is reading Application Notes from each supplier.
Regional differences matter for unit protection vs grid quality in your choices.
A good line filter is best start ( All earthed SMPS have some sort of line filter)
Outside signals vs power lines is another condition where above ground distribution is more exposed to lightning impulses from stray nearby hits. Unlike our residential areas where all pnone CATV and power cables are underground with shielded XLPE coax for power.