Vice

Refers to the one second in command, next in authority to another person. It's used for people who are one step away to the highest authority. Their job is different from the higher person's. They're not authorized to act on behalf of the boss, unless they're told to do so.

Usually, a vice is someone, a person (maybe two persons, but not more). Not a group of people. Example:

vice president, vice admiral, vice chairman

Meanwhile:

Deputy

Refers to the one(s) appointed to act for another person (not necessarily next in command), to take their place on their behalf. What they do is subset to the higher person's authority. Usually, a person can have a number of deputies. Example:

Deputy manager, deputy sheriff.

The word deputy is also commonly used in organizations. At where I work, we have six deputies (not six vices).