For getting up close and personal to the falls, the Argentine side is a lot better. There are two routes: the paseo superior and the paseo inferior. They both are a series of trails, bridges and catwalks that give you spectacular views of the falls. The paseo superior goes over the top, while the inferior climbs down to the base of the falls, so you can get a good view looking up.
Another advantage to the Argentine side are the two free boat tours that will get you close to the falls as well. It seems the locals will do anything to get you wet.