Between the two, I would say that John Williams is the better. He has had more great pieces over a longer and more consistent career. Hans Zimmer did a lot of good work when he got started, but once Gladiator came out, something changed about the way he wrote his music that I haven't cared for. I'm not sure exactly how to articulate it, but the closest I can come is to say that his music now seems stale. maybe it's just me.
But, please don't limit the list to just those two. Many great composers have graced the speakers next to the screen with wonderful music. Here are some examples
Jerry Goldsmith is an absolute MUST for this list. The man could compose a great tune in his sleep. His main theme to Star Trek is Immortal, and his scores have consistently made movies something more special than any director could hope for (i.e. Air Force One and Rudy.)
James Horner, like Zimmer had a lot of really good hits early on and eventually tailed off, but you can't deny he has made good contributions. I personally think that he needs to get out of the rut of the modern-pop-orchestral score. It's great for Land Before Time, but I still prefer Rocketeer to Titanic.
Basil Poledouris has composed some really great music. Often times his music is hidden by either a ridiculously made movie (what I think of Starship Troopers) or a really well made movie (what I think of Robocop.) I think that the best balance between marvelous movie and marvelous score came with is work on Hunt for Red October (likely his most widely known score.)
Bernard Herrmann.... What's there to say? Very artistic and versatile composer. Has made a profound impact on a lot of current greats.
Danny Elfman needs to be on the list just because of how he shaped the little quirks of the modern industry. First, he created a signature "quirky" sound that is often requested by many directors. Second, the signature Heroic sound that he showed in Batman not only rescued the audience from having to sit through an hour and a half of prince (he is great, but one can only handle so much/ and YES, Prince was originally going to produce songs that would score the entire film) but would also set the bar by which comic-book super-hero scores written after were judged. Not to mention that, after Batman, all the big composers started using concert pitch scores just because "That's what Danny did with Batman" (and that is one of his orchestrators talking. I'm not just pulling that out of my ****
Anyway, moving on....
Howard Shore has obviously been vaulted into stardom by the LOTR trilogy and what I've heard of his music is somewhat a mix of Goldsmith, Zimmer, and Horner. I'm also interested in hearing what comes of his recent venture into the realm of video game music.
John Debney has shown the ability to adapt to many film genres from Sci Fi (Zathura) to Americana (Last Race from Dreamer) to massively Religious (Passion.) Personally, I think he's on track to be a solid A-lister. And he's an insanely nice guy. I was lucky enough to meet him about 2 years ago. He is very accessable and seems willing to help youngsters who want to get started.
James Newton Howard is likely one of the most versatile composers on the planet. He can reproduce the John Williams sound like it's second nature, his sound by itself is very interesting and can keep listeners hooked, and he has roots in smaller scale scores (like Major League) so he can make a lot out of a little.
I'm very sorry if I've sounded as if I'm writing PR notes about some of these composers. I'm also sorry if this is a long and horrific read. This is just my perception of them and not meant to be taken as bible at all. Hopefully they're good food for thought.