The only thing that changes is that the 2nd one to go up in the air will get a letter added to its call sign, to distinguish the two. So if, say UA 936, an Airbus 320 DEN-IAD is delayed, then the continuation IAD-ZRH on a 767 may be labelled as 936a. On Ch9 I've also heard ATC assign the first letter of the pilot's name as the distinguishing letter.

Edit: I should not that I _think_ but am not sure that the 2nd flight is the one that gets the letter added.