|The following is based on the canceled Fallout movie and has not been confirmed by canon sources.|
This was meant to be seen as a prestigious job, or at least designed to appear as such. The Assistant Vault Supervisor was seemingly responsible for the supply inventory. Jobs that are revealed in the film that this role involves include checking the Garden of Eden Creation Kit and the vault's water supply.
The fact Hero, the current incumbent during the film's beginning, makes a presentation on the water supply's defect to his father, the current Vault Supervisor, suggests that the Assistant Vault Supervisor is subordinate to the latter, as does the name.