Lord of the Rings is really not aimed at children (The Hobbit was) the way Potter is, so the movie audiences will be skewed differently. Anyone can go to Potter, but I'll be surprised if there are a lot of young children at Rings. It's a more in-depth story, with more seriousness, more violence, more textured reasons for evil and good -- none of which will appeal to the 7-year-old set that can't get enough of Harry.
Although adults can enjoy Harry Potter books and movies, it isn't primarily directed at them. The Rings books and movies are very much directed at teens-on-up, and have a built-in audience of people who have waited up to fifty years to see this on film. That's two or three generations of built-in audience, just aching to see this movie.
Given the way the audience is slowing down for Potter already, and given that I've read a rave review of Rings in Newsweek and Rolling Stone called Rings the best movie of the year, I'll bet Rings gives ol' Harry a run for his money. Both will be megahits, but Rings was made slightly outside the Hollywood system, and I'll bet it will show in the depth of the storytelling. Depth sells to the adult audience.
Just my guess. At any rate, it's nice to have a couple of great movies out this season that everyone (or almost everyone in the case of Rings) can enjoy.