There's no standardized answer for that, to the best of my knowledge. Whatever the company wants to go with is fine, really. If we really want to get into the specifics, a deputy director would be chosen (aka deputized) by the director, while a vice director may or may not have been elected by the board of directors. However, the board of directors could have simply chosen the second director and then named him the deputy director on their own. Again, it's a matter of personal customs and decisions; they could also call the man the assistant director, the co-director, the secondary director, or even the junior director. Hell, they could name him King of Directorship if they choose... That all being said, "Vice X" is the most COMMON title for a person in any second-in-command administrative position, so that's the one you'd be most likely to hear.