I need to write a simple 3D billiard game using OpenGL. Which physics engine is simple and best to use in this case? Or it's better to write physics by myself?
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I've written a simple 2D billiard-like game using Box2D before.
As Digital Architect pointed out, unless the ball jumps from the table, 3D physics is overkill. And you can use 2D physics with 3D graphics.
If you decided to implement physics by yourself, this is what you'd have to do (which are already implemented in Box2D):
Also, you'd have to make sure everything works properly (no obvious bugs, which can be a pretty hard problem unless you really know what you are doing) and efficiently.