USE: When the series resembles that of a random walk, taking first differences makes it stationary, so that it can be described as linear series representation of autoregressive or moving average terms.
DO NOT USE: When the series appears to randomly fluctuate around its mean.
USE: Similar to simple differencing but applied when the variance in the series is assumed to depend on the level. Or when the series is an index.
DO NOT USE: When there's no such assumption or where the series is not strictly positive.
In classical time series decomposition you should proceed as follows:
Decide on whether taking logs of original series is reasonable
Decide on whether regression on time variable is reasonable
Decide on whether to take differences of the residuals