A Game Engine is the core software component of a video game. It typically handles rendering and other necessary technology, but might also handle additional tasks such as game AI, collision detection between game objects, etc. The most common element that a game engine provides is graphics rendering facilities (2D or 3D).
From : en.wikipedia.org/wiki/Game_engine
Gaming Middleware developers attempt to "pre-invent the wheel" by developing robust software suites which include many elements a game developer may need to build a game. Most middleware programs provide facilities that ease development, such as graphics, sound, physics and AI functions.
From : http://en.wikipedia.org/wiki/Game_engine#Middleware