For those who is unfamiliar with Compiz, let me introduce it first. According to this site, Compiz is an OpenGL compositing manager that use GLX_EXT_texture_from_pixmap for binding redirected top-level windows to texture objects. In other words, Compiz is an open-source equivalent of Microsoft's Aero.
I have made some comparison between Linux's Compiz vs Windows' Aero. 0.5 points is awarded when one beats another in appearance, and 1.0 points is awarded when one beats another in functionality.
Firstly, they differ in their appearances. Compiz has many special effects that Aero does not. One example is the wobbly effect. Wobbly effect is the main reason that my friend V. T. installed Linux in her laptop (although she could not get it working). And it comes with plugins. Even I can use it to embed a terminal (command line) input in my desktop wallpaper. My friends, even my biology teacher, amazed at my laptop's extraordinary look. Compiz beats Aero in appearance. 0.5 points for Compiz.
Score: 0.5 - 0.0
They also greatly differ in customizability. Besides having many special effects, in Compiz I can turn off effects that I do not want, and turn on effects I want. It also accepts plug-ins. Sadly, I cannot add any plug-in in Aero, nor I can program any window rules in Aero. In Compiz I can use it to embed a terminal on my wallpaper, mentioned above. I also can set the spring constant to model window oscillations in the aforementioned wobbly effect. 1.0 points for Compiz.
Score: 1.5 - 0.0
But, without proper tweaking, Compiz can show clumsy desktops. Provided that you have a fast-enough computer, Aero, always show the elegance and the beauty of glasslike windows, no matter what. 1.0 points for Aero.
Score: 1.5 - 1.0
Wait, have you experienced that Aero is a memory-consuming giant? Aero is responsible for making V. T.'s laptop so slow when running Windows. But, with Compiz, this is a completely different story. 1.0 points for Compiz.
Score: 2.5 - 1.0
What about hardware support? Linux (and therefore Compiz) supports fewer graphics cards than Windows. This is the reason why V. T. could not get it running on her aforementioned laptop. 1.0 points for Aero.
Score: 2.5 - 2.0
Compiz pwns Aero when it comes to customizability and features, but, remember that in reality Microsoft still pwns Linux in terms of hardware support. Aero excels in its default elegance and hardware support.