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"a game engine is the base element which strings all aspects of a game into a single package which can then be used to create your title."
that was said at SIGGRAPH back in '96 when someone first asked why DirectX was an API and not an engine, because at the time it did appear alot like an engine.
Generally the setup goes
Game Content -> GameEngine -> Audio Engine -> Audio APi
.......................... -> Graphics Engine -> Graphics APi
.......................... -> Input Engine -> Input APi
.......................... -> Network Engine -> Network APi
if you follow sports at all, you could think of the seperate part ofs the engine as the players ... the GameEngine itself would be the coach/manager on the sidelines telling them what to do and where to play, in order for them to score the game content properly
the APi itself would therefor be the skill they have to accomplish the said tasks