Honestly, there are many f/x CGI studios out there that do work for movies, but unless it's either of those two it's like fanboys don't give a rat's ass about it. Rhythm and Hues did this Hulk, and I have seen their work before in many a movie, and they're not slouches in the industry, neither are Sony Imageworks. I think unless we don't hear who is doing the CG suddenly we call it shoddy work or sub-par. Weta and ILM have both done extremely well work, but they have also done some ****ty work.
Well ILM and WETA are the biggest ones definately. I guess the success of WETA comes with their recent wins of the VFX Oscars ( LOTR and King Kong). But i think that each VFX studio has their own experience. I'll list some of the bigger studios and what i know of their work
WETA's experience is definately performance capture. Gollum , King KOng are the most famous examples and because of the performance capture experience James Cameron chose them as the VFX firm for Avatar.
But consider that Cameron himself has developed his own VFX technology for many years and WETA is the one which will use it. WETA also seems to be one of those few studios that can perform really difficult CGI in a short space of time. They also worked X3 and regardless what what you think of the movie , you gotta admit that those VFX were great.
Sony Image Works specialises in virtual humans. In the last few years many they have been the ones who have constantly been at the forefront. Spider-man 2 and 3 , Superman , Beowulf , MOnster House and POlar Express. AS well as virtual cities . I remember watching the VFX featurette of Mtrix Reloaded and the VFx supervisor ( John Gaeta) mentioned that they have some really good shaders for producing atmospheric CG shots. The problem i sometimes have with their work is that although they have very realistic CG humans , they stil'l haven't reached that level where it should be. As for as creature animation goes , it is a hit or miss. With Spiderman 3 they really delivered excellent work with Sandman but in I AM LEGEND the vampires looked awful sometimes.
Rhythm and Hues specialses in realistic creature animation and to me knowledge , Hulk will be the first movie where they will be creating a CG human. But yeah creature animation or more specifically animation of animals is what they usually are sought after and they do deliver some very good work. Look at their latest movies :
Alvin And The Chipmunks
The Golden Compass
Charlotte's Web
First Narnia movie
Tippett Studios like R&H is also sought after for creature animation but they tend to go for the more "fantasy" based creatures then real creatures. They often are used for smaller pictures but they did loads of work for the 3rd matrix movie , constantine and let's not forget the bugs in Starship Troopers.
And of course there is ILM. ILM is the the biggest firm and have produced many of the famous VFX shots of classic movies ( Jurassic Park , Terminator). I think up to the 90's they were the to go VFX firm ( and probably still are) but with the rise of the other VFX firms they faced huge competition.
You can also see that with the OScars they won. Although they have been nominated each other year , they finally won an OScar again for their work in POTC 2 after their last win '94 for Forest Gump.
'95 was Babe , '96 Independance Day , '97 Titanic , '98 What Dreams May Come , '99 Matrix , '00 Gladiator , '01 FOTR , '02 TTT , '03 ROTK , '04 Spidey 2 , '05 King Kong , '06 POTC 2
ILM has "made a comeback" ( if those are the right words) recently with their photorealistic animation of Davey Jones and Transformers. And i's not like they also have had the great VFX . Remember Van Helsing and the second mummy movie.
There are other big studios such as Digital Domain but honestly i don't know what they specialise in.
Fanboyism plays a big role in determining why many people want ILM and WETA . Especially the case with WETA because of their recent oscars wins. But if you look closer alot of those studios really do deliver some excellent work. VFx isn't just about animating something realistically . It's making something believable. Sometimes that costs loads of money and sometimes it just requires the right talents. And studios always are bringing their own VFX techniques to their last projects.