Well I cannot say what's common in USA, but the literal meaning of cash seems to me to be getting literal cash in your hand. While deposit is to deposit it into an account, for withdrawal later, like when the check is cleared.
I'm guessing if people use those terms interchangeably, they are in error half the time.
IDK practices in USA, but here some checks can be cashed immediately when they are from trusted parties, like a bankers check, but AFAIK you still need to have an account with the bank to cash it.
That said both cash and checks are extremely rare here now, almost everything and everybody have gone electronic.
Paying cash above about 8K USD or 50k DKK is actually illegal now, to prevent money laundering.