I've heard the phrase "baritone euphonium" but I honestly don't know what it refers to, I've always assumed "baritone" in that phrase was a description of the voicing the instrument the "euphonium" is playing/singing. It could be something the yanks say, I'm pretty certain they have their own nomenclature.
In the picture above, there are two distinctly different instruments. A Euphonium with a conical bore and a baritone with a cylindrical bore. This is similar as to how the trumpet (cylindrical) and cornet (conical) are completely different instruments. A conical bore produces a more "mellow" tone.
With reference to the picture, here are some differences I think I see:
- The Euphonium have 4 values vs the 3 on the baritone.
- Notice the length of tubing above the eupho players fingers, the baritone lacks that because it can accommodate tighter turns in the tubing due to the cylindrical bore.