Real wood looks much better since it is a natural material. It can help keep pH lower, and some of my pleco species need real wood to munch on.
There is some likelihood that real wood can also harbor actual de-nitrifying bacteria in an anaerobic zone deep inside the wood. These would be bacteria that actually convert nitrates into atmospheric nitrogen, actually removing nitrogenous wastes from the tank.
It is also fun to go hunting for driftwood out in nature too.
But wood does require more work since we often collect it ourselves and have to boil it to kill any weird stuff that might be living on it. It can also take some time to fully sink, and it can stain the water yellow due to leaching of tannins.
I also use the wood as a substrate for growing Java Fern and Anubias. I don't know if the roots would cling onto plastic quite so well.
Fake wood just looks too fake to me. I work hard to create a natural environment for my fish, so I just can't see ruining that by adding fake stuff. However, it would be guaranteed clean, easy to sink, and easy to clean. It would never rot. So I guess there are good things about it.