So would this be a more-or-less correct tl;dr:

Dimensionality reduction is a compressing data in a way that retains the most important information for the task

Feature selection is removing unimportant information (keeping/collecting, or selecting, only the important parts)

Both cut down on the amount of data you end up with, but one does it by finding a representation that is smaller, the other does it by discarding unnecessary data (or, rather, telling you which data is necessary, so you can stop collecting the unnecessary data).