From an accounting point of view, a debt is something that you owe because you have borrowed. A liability is anything that you owe or will owe. A debt is a liability, but not all liabilities are debts. For example, if a business owes taxes, that is a liability. It is not a debt.
'liability' can also be used outside of accounting to simply mean anything that is a negative.
He's brilliant but his tendency to drink too much is a liability.