Some of the Factual Differences between Baritone and Euphonium:   
(the picture is a euphonium)              
A Euphonium has mostly conical tubing, the Baritone is mostly cylindrical.
A Euphonium has a larger bore than a Baritone. 
The Euphonium often uses a deeper-cup mouthpiece, resulting in a darker sound than the shallower mouthpieces usually used by Baritones.