All right....
Other than the mouthpiece and receiver being shorter on a cornet than on a trumpet and the shank taper being smaller, too (although the cup, throat, backbore, and leadpipe taper are still essentially the same) a trumpet is longer than a cornet, which also has more crooks (unless it's a Conn Connstellation or an Olds Mendez, which look exactly like their trumpet counterparts), except for a pocket trumpet, which is much shorter than a cornet and has even more crooks (unless it's a pocket cornet, in which case there is no difference in appearance between the two [like the visual similarity between the Conn Connstellation and Olds Mendez mentioned above]), and as for timbre and projection, a trumpet and a cornet can both sound the same or different, depending upon the player, the mouthpiece, and the design and construction of the instrument.
Whew! There - the difference between trumpet and cornet in a (coconut) nutshell. Got it?