GLUT isn’t really a graphics engine. GLUT is a windowing/input library. That is, GLUT provides a way to put a window up on the screen and receive events from the mouse, keyboard, etc. OpenGL is what you would actually use for graphics with GLUT or the other libraries mentioned on this site. OpenGL provides a C API for doing 2d/3d graphics, but there are bindings to many other languages, including C#.
I would not use GLUT for games, as it is rather limited. SDL is a better library that is more full featured and gets highly reccommended by those who use it. SDL is in C, but again has bindings to other languages.
I don’t actually know anything about C# XNA, so someone else will have to compare them for you, though I will say that OpenGL is fairly low level so there may be functionality in C# XNA that you would have to implement yourself in OpenGL.