"en même temps" vs. "à la fois"

Yes, they are very close and, often time, can be used interchangeable. But there are cases where one can't substitute for the other. The safe bet is to use en même temps if you want to say that somethings are happening simultaneously, connected to a concrete action:

Ils sont arrivés en même temps. -- They arrived at the same time.

 

À la fois has acquired a broader meaning. It still is "at the same time" but not necessarily referring to clock time. Often it is best translated as "at once".

Une personne à la fois. -- One person at a time.
Marie est à la fois jeune et belle. -- Marie is at the same time young and beautiful.