Balancing means each tire is taken off and spun on a machine which measures where to put small weights so that it is ‘balanced’, i.e. that its weight is evenly distributed along its rotational axis (in the old days this was done ‘static’ i.e. w/o spinning it, just balancing it horizontally on a sort of sensitive mounting shaft).
Rotating means all four tires (again, in the old days, five, including the spare) are removed and their position on the vehicle is rotated (fronts to back, backs to front, as well as left to right or whatever ancient chinese secret your mechanic believes in ;)).
I personally don’t believe balancing needs to be done unless something is done to it (other than simply taking it off the car and putting it back on), i.e. a new tire is put on the rim, its plugged or patched, or when they’re rotated (of if they aren’t riding smoothly).