Game Library?

At the very least, game libraries like Allegro will provide functions that make drawing easier. They also tend to have other functions that aren't related to graphics but make game development easier. Note, for example, this snippet from the unofficial Alternative Game Libraries FAQ:

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From the AGL FAQ [Allegro] provides many functions for graphics, sounds, player input (keyboard, mouse and joystick) and timers. It also provides fixed and floating point mathematical functions, 3d functions, file management functions, compressed datafile and a GUI.

Also note that OpenGL or Direct3D can be used with Allegro, along with a host of other options as mentioned on the Allegro introduction page.

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What does a Game Programming Library like Allegro has to offer over a Graphics Library like OpenGL?

OpenGL won't help you play music or read from the keyboard.

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I may be wrong here, but doesn't allegro use OpenGL for its rendering?

No, under Windows it uses DirectX. But AllegroGL does, as its name implies, use OpenGL for rendering.

The main thing that a library like Allegro gives you is a cross platform way to easily access hardware commonly needed in creating games.

Allegro uses OpenGL (probably DX as well when compiled on a windows machine like SDL) for its rendering. This doesnt mean you can simply use OpenGL and have all functionality of Allegro!

Allegro offers much more than just graphics (like sound and input), and is also much easier to use. Have fun learning how to use OpenGL for your game while you can also use some game engine that will do the job 10 times faster!

Plain Allegro never uses OpenGL.