The difference between subtracting the mean and dividing by the mean is the difference between subtraction and division; presumably you are not really asking about the mathematics. There is no mystery here, as it's no more than a statistical analogue of
with the difference that the mean is used as a reference level, rather than another value. We should emphasise that
- (Bill $-$ Betty) or (value $-$ mean) preserves units of measurement
- (Bill / Bob) or (value / mean) is independent of units of measurement.
and that subtraction of the mean is always possible, while division by the mean usually
only makes sense if the mean is guaranteed to be positive (or more widely that no two values have different signs and the mean cannot be zero).
Taking it further then (value $-$ mean) / SD is scaling by the standard deviation and so again produces a measure independent of units of measurement, and also of the variability of the variable. It's always possible so long as the SD is positive, which does not bite. (If the SD were zero then every value is the same, and detailed summary is easy without any of these devices.) This kind of rescaling is often called standardization, although it is also true that that term too is overloaded.
Note that subtraction of the mean (without or with division by SD) is just a change of units, so distribution plots and time series plots (which you ask about) look just the same before and after; the numeric axis labels will differ, but the shape is preserved.
The choice is usually substantive rather than strictly statistical, so that it is question of which kind of adjustment is a helpful simplification, or indeed whether that is so.
I'll add that your question points up in reverse a point often made on this forum that asking about normalization is futile unless a precise definition is offered; in fact, that are even more meanings in use than those you mentioned.
The OP's context of space-time data is immaterial here; the principles apply regardless of whether you have temporal, spatial or spatial-temporal data.