What is the difference between "safe" and "vault"?

Solution 1:

A safe connotes something smaller than a vault, and may be movable, while a vault is generally built in-place and would be torn down rather than moved.

The computer jargon use of vault may be due to its connotation of greater security. It is also attractive because it is less common, and is only a noun (in this sense) and thus quicker to interpret, whereas safe can be noun, verb, or adjective (and heavily used in the latter sense in the computer security context).

Solution 2:

A safe is usually smaller than a vault. Safes are often portable, or at least movable, while vaults often are entire rooms.