There are plenty of direct sequels that are pretty damn good, if not far superior to the original. However, someone mentioned it before, by the time the third movie rolls around, their is little room for improvment. Only two third movies were superior to the previous two.
Here are some movies I know will be brought up, but still arguable as to whether they were better or not.
Aliens: James Cameron takes us on this helluva ride back to LV426. We get introduced to Colonial Marines and the Alien Queen. Was it better then the original. Ask anyone over the age of 30, and they will tell you know. I'm on the fence about it. I enjoy the action that was this second film, the sheer terror of the first movie and not knowing what the creature was is amazing.
Terminator 2: Again, when people mention sequels that were better, this one comes up almost instantly. Visually, this movie is a delight, and the T-1000 is an amazing creature. James Cameron delivers again in another action packed thrilled movie. However, is it better then the original. Terminator was the first movie to really explore time travel and screwing with time. The love story was pretty damn strong and Michael Beihn played the lead role very well (very unsung actor, even as Hicks in Aliens). Arnie was on as the evil emotionless Terminator, which I think he played much better in this movie then the cuddly kid friendly Terminator in the 2nd film.
Spiderman 2: For some reason, this movie gets a ton of attention,a nd personally, I can't stand it. Doc Ock is about as boring of a bad guy as I've ever seen ont he big screen. Spiderman is just a whiny bitch all throughout this movie. The first movie, while not great, is much better. If you want to talk about an over rated movie series, then look no further then Sam Raimi's adaptations ofthe Webslinger.
Sequels I would say are better.
Scream 2: This movie was a movie about sequels, and pretty much got the forumula done right. The first movie was fun, but flawed. This movie was pretty damn smart, and got it right on concerning third movies, they suck, much like Scream 3 did.
The Two Towers: An amazing movie. Fellowship of the Ring is seen as probably the best of the three, but 45 minutes focusing in the Shire and the Hobbit is much too boring and slow to my taste.
Empire Strikes Back: When it comes to sequels, this is the formula in which all movies should try to make. While Star Wars is a visual treat, it's very forumlaic as far as basic story telling goes. It retells the same story in a different matter, and it worked. How could Empire possibly live up to the hype and hoolah surrounding what was the biggest movie to ever come out?
A change of director's with Irvin Kirschner sealed the deal. Kirsch switched it from a huge special effects movie, to a very personal, and emotional character driven movie. Gone was the large space battles, but more of the personal conflict. This movie is by far the best sequel ever, and probably left us with the best line in cinema history.
Third Movies that are the best:
Return of the King and Revenge of the Sith (If you break the Star wars movies into two trilogy that is.)
There were plenty of sequels that ruined the legacy of a film series.
Look no further then the Matrix Sequels, which turned the Matrix (which was touted as one of the best and most original movies of the 90's) and effectively turned it into a running joke. Batman was killed by Forever and Robin.
I could go on for ever though.