This event seems to be the singular instance of this happening in all the time travel involved in the Flash, so it's likely that this was just creative licence, because trying to deal with a second Barry in this episode would have been too complicated on the writers' part.

However, if you want an in-universe explanation, I would say this: 

The "other" Barry is from a future where Barry went back in time, but because our Barry saw him, it led him on a path where he ended up going back in time (where otherwise he wouldn't have done) to create a past where he didn't go back in time, causing the other Barry to be erased. When he later became the "other Barry" the positions were reversed and the "original" Barry disappeared because he was from an aborted timeline. To elaborate on this, when Future Barry went back in time and was seen by his earlier self, he disappeared because he was transported immediately to HIS version of events that occurred as a result of the original Barry going back in time. To put it a simpler way, because Barry's time travelling was a predetermined event from the beginning, the events we see from the moment near the beginning of Out of Time and the moment he leaps at the end, are THEMSELVES the alternate events that only Barry will ever have experienced. In a sense, during this episode, Barry is truly "Out of Time".