In terms of dinosaurs: a century ago they were the two prominent museums in the US competing directly for specimens. (They still are of course but there's more competition.)
I'm biased, but I feel like the Carnegie does better with both the old-timey early 20th-century prehistoric displays, and also the modernized mounts and scientific story-of-vertebrate-evolution. It's close. High marks to both.
Note, if you want tyrannosaurs, don't miss the Museum of Rockies in #bozeman