There's also the cohomology version of Ryan's answer: the Leray-Serre spectral sequence, which tells you some very nice things about the cohomology of a bundle, and essentially nothing useful about the cohomology of a submersion. You can consider this a particular instance of Tim's comment.
In general, algebraic geometers and homotopy theorists work with bundles (or more generally, fibrationsfibrations), every day of their lives, and will extremely rarely encounter submersions. Even if you don't want to work in such fields, their existence is a good reason to distinguish bundles from submersions.
