Personally, I have more reservations with a young child watching the Harry Potter series than I do LOTR. But that's because I believe themes like "There is no good and evil, there is only power," are much more dangerous to young minds than a little bit of blood and gore. At least with LOTR, there is no blurring of the lines between good and evil.
That's what I think made the original Star Wars trilogy as good as it was. The Prequels blurred the line between good and evil quite a bit and I think the story suffered for it.
That's just my opinion though. If I had kids of my own, I would definitely allow them to watch LOTR, but I wouldn't necessarily want them to see Harry Potter until we had a long talk about the subject matter of that story.
Have you seen (or perhaps more importantly, read) all of the Harry Potter series? I ask because I'm not sure where you're getting the idea that they promote that "there is no good and evil, only power." There are still pretty strong distinctions between good and evil in the story. It's not always clear where some characters fit on that spectrum (but it's usually revealed in the end), and Harry worries about turning evil (which is a pretty common thing in stories like these - see Luke in ROTJ), but the series is clear that people are not born inherently good or bad but that their choices define them. Without verging into Rancor Pit territory, I know that extremely religious people have taken issue with the magic in the books (as well as things like the disobedience of authority and, presumably, the characters that don't fit a good-or-evil mold) but most people are able to see the good things that the books actually promote, like love and friendship being the greatest powers of all. (For the record, I've always been a much bigger fan of HP than LOTR - the books are much more fun to read and the movies are much more fun to watch, quite frankly.)