The Vault 77 jumpsuit gives +5 to melee weapons and unarmed skills, as it is said that these were the Puppet Man's favorite methods of murder. As there was only one inhabitant of Vault 77, and as Vault 77 isn't in the game, the suit is unique, like the modified utility jumpsuit.
When you wear the Vault 77 jumpsuit in Paradise Falls or around any other slavers, it has no effect on them despite the fear of the Puppet Man in slaver colonies and the "Burn This Goddamn Jumpsuit" holotape.
The suit is tagged as the slavers' property, so taking it is considered stealing.
The Pip-Boy icon for the jumpsuit's filename is " "vault_suit_76.dds."