I have to agree with a lot of what
Beanamir already said. Comparing Shore and Zimmer based on just one production feels unfair seeing that they both have composed numerous soundtracks for very different kind of films.
I don't know much about Shore's other works besides LotR, but to make it possible to compare him and Zimmer, maybe the question could also be about which of these two composers is better at adventure/ war films.
In LotR, Shore did amazing job creating music for different kind of places and people from Hobbiton to Mordor, west to east, which undoubtedly required a vivid imagination. But hasn't Zimmer done the same, taking us from the Caribbean seas to a 19th century Japan (The Last Samurai) and the Roman empire in between (King Arthur, Gladiator)? One may think whatever about these aforementioned movies, but personally I liked their soundtracks a lot, and in many cases the music did a lot to improve the story.
If we are to separate the score from the films, I think I like Zimmer a bit better. Shore's music goes together seamlessly with the events on screen and that's an essential skill for a movie composer, but Zimmer's music can stand alone; it's easy to listen to without any context. However, if we talk about the music as a part of a film, Shore's versatility within one movie (even if it was a 11-hour-long one) is just awe-inspiring whereas while having some true gems in his scores, Zimmer sometimes seems to get stuck with a certain sound or a "feel" for an entire soundtrack. But should we blame the movie or the composer for that?
(Okay, so it is a cop-out to end a post in a question hoping that no one notices that I didn't really answer the original inquiry, but I think I could pick whichever and be happy with my choice.
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