When it comes to average hatchery/feedstore based stock there isn't much difference at all except in comb type, color, and individual bird personality.
Breed Standard-wise, Cochins should be rounder like a basketball with a head, with a very prominent, round rump and hardly much of a tail to speak of, with a more subtle facial expression, with much more leg/toe feathering, and of a slightly lighter weight. They will often appear to have very short legs, bantams especially. Brahmas should have a wider skull, more serious expression from wide brows and deeper set eyes, and a more smooth (not so round), slightly upward sweeping rump. They should have higher wing carriage, less fluff, and more noticeable, strong legs and less leg/toe feathering. The males also have higher set, more noticeable tails.
Cochins often have a more "proper and show" type presence, Brahmas have a more "proud and serious" presence.
The other two differences are Cochins having a single comb, Brahmas a pea comb.
Both lay a good amount of large light brown eggs, both can be often broody if from the right strains, both are cold hardy and good, meaty dual purpose birds, but only if from the right strains.
I've never had Cochins before except some hatchery/layer type bantams, but I've got both hatchery and rare-colored/imported type Brahmas. Their egg color ranges from light brown to a chalky deep pink.