Ease bought by someone having done work in advance. Convenience is never free; it was paid for earlier.
Latin convenientia (agreement, fitness), from convenire (to come together, suit), from com- (together) + venire (to come); ease that arrives only because someone did the coming-together in advance.
Ease bought by someone having already done the work of arranging things to fit. It is never free — it was paid for earlier, by someone else's effort.