Was the third LOTR film WAY better than the first two or were the films in 2001 and 2002 that much stronger such that LOTR didn't win major nominations for films produced those years?
It is a fair question. There has long been a Hollywood bias against SF/Horror/Fantasy genre pictures. Back in 1986 Dennis Hopper was heavily talked up for Blue Velvet. He got nominated for a smaller role in Hoosiers instead. He had cleaned his life up and it was a pat on the back. Frank Booth was too unfriendly a character to be nominated "best" anything. Fast forward a few years and Anthony Hopkins wins for a role as a cannibal serial killer in Silence Of The Lambs.
Times change. Some in Hollywood detest that sort of film (and not just because of the grue). Why should "old Hollywood" types be silenced in their championing a different film?
Meanwhile there are still attempts to revoke Michael Moore's Oscar although I don't ever see that happening.
When Return of the King came around, somehow the series had gone beyond the preconceptions that the Academy has about fantasy. And the Academy has a complex about rewarding things that are "due". The fact the Return of the King was the third, and widely viewed as the best, in an acclaimed trilogy, and the fact that the first two films were Best Picture nominees but lost helped seal its win. It was pretty unbeatable.
In other words, people sort of rallied around Return of the King because the Trilogy was due, but it also was more worthy of winning. For example, in 2003 there was a BIG campaign by Harvey Weinstein for the Academy to vote for Martin Scorsese for Best Director because Martin had never won before, not because he deserved it that year. The overzealous campaign backfired, and Scorsese lost again.
So as for ROTK, they didn't just vote to reward the series. If ROTK wasn't good enough to justify the win, it wouldn't have won, though guess is that the "due" factor sort of overrode the Acadamy tradition of snubbing fantasy/sci-fi films.