Plants provide a much healthier enviroment for fish. They take up stuff that would otherwise be toxic to your fish, (also help with the biological balance of things going on in the aquarium). Plants also help relax and bring out colors in fish and give them a place to hide and recoup when they are scared (although so don't fake plants). However, IMO, live plants make an aquarium look SOOOO much nicer than fake plants. I actually think they are easier because plastic plants, IME, gather algae over the long period, it becomes hideous and then is very hard remove.
They also give you the benefit of hearing "Oh, I love your tank!"

(not all the time though).
The down side of plants is that they are actually alive. You have to clean away dead material from them, provide light and nutrients for them etc, etc. If you don't remove dead stuff, then they rot and add excess ammonia to the water, which may hurt your fish. Some plants don't grow roots, like hornwart, and come easily up out of your gravel or sand... others are just plan difficult to keep down.
However, IMO, the good way out weighs the bad and live plants make an aquarium look SOOO much nicer.