Both "one after the other" and "one after another" are used, with no real difference in meaning.  Having said that, I will make a the following observations:

First, if there are only two objects, I would be inclined to say, "one after the other."   Example:  "I had two books, and I read one after the other."   "one after another" creates a feeling that there are three or more objects.  "I had twelve books, and I read them one after another."  The more objects there are, the more I would be inclined to say "one after another."

Second, "one after another" can create the impression that the events are happening very quickly.  That is, there is very little time between one event and the next event.  "He ate five bowls of ice cream, one after another."  This makes me think that as soon as he finished one bowl of ice cream, he started the next.  Compare:  "He ate five bowls of ice cream, one after the other."  It means the same thing, but somehow I don't imagine him starting a new bowl of ice cream quite as fast.