As with so many questions, the answer is, "It depends." :-)
Here's my understanding--in British brass bands, they are two distinct instruments. The big difference is in the degree to which they tend to be "cylindrical" or "conical" in their diameter from mouthpiece to bell. A cylindrical-bore instrument (such as Trumpet or Trombone) keeps the same diameter throughout much of its length before it reaches the bell flare; a conical-bore instrument (such as horn or tuba) expands in diameter gradually throughout much of its length.
The cylindrical-bore instruments have a brighter, more edgy sound, while the conical-bore instruments sound more mellow. Back to your question: the British "baritone" is a cylindrical-bore instrument, and the British "euphonium" is a conical-bore instrument (also usually with a bigger bore to begin with). In British brass band music, the baritone tends to be used more as a solo instrument, where the euphonium tends to be kept paired with the tubas. Other than that, they are usually pitched in the same key, and they finger the same. The trumpet and cornet are perhaps a more familiar example of the same thing--the trumpet is cylindrical-bore, where the cornet is conical-bore, thus accounting for the different sounds.
Here's the catch--in the U.S., a lot of us use the terms "baritone" and "euphonium" interchangeably, probably because U.S. bands tend not to use the British "baritone" at all. (We don't have this kind of confusion with the terms "trumpet" and "cornet" because both are familiar in the U.S.) I usually call our conical-bore instruments "euphoniums", as the British do, since that seems to be the best way to avoid confusion.
I've attached links to Besson's web site where you can see the difference between the British instruments.
A baritone horn is slightly smaller than a euphonium. It is closer in size to alto/tenor horn, and therefore the Euphonium has a darker tone. The Baritone and Euphonium have the same pitch, same fingerings and read the same music, they just produce differing tones. Euphonium also tend to have a wider bell. The main distinguishing factor in America is a Euphonium has a conical bore (the pipe gets larger as it goes through the horn) where as a baritone has a cylindrical Bore (the pipe stays the same diameter all the way through the instrument, until it reaches the bell). This give a Euphonium a mellower sound than a baritone.
Baritone horn, Sousaphone, Euphonium, Contrabass bugle, and Alto Horn
The euphonium is also known as the tenor tuba, baryton(Ger.), basse à pistons(Fr.), and sometimes mistakenly referred to as a baritone horn or just a baritone. The baritone horn, however, is a cylindrical bore instrument with a brighter tone (closer to trombone timbre) than a euphonium. The bore of the baritone is often much smaller than that of the euphonium. The euphonium is a conical bore (gets larger is we reach the end) instrument with a warmer and darker tone than a baritone.
French Horn Tuba Trumpet Baritone Euphonium Trombone Cornet
Trumpet, French horn, trombone, tuba, in an orchestra; euphonium and baritone horn in a concert band; tenor horn and alto horn in a British-style brass band.
If the pipes of a baritone were straightened out it would be 9 foot (2.74mtrs) end to end, which is the same as a trombone and a euphonium.
Baritone horn, Sousaphone, Euphonium, Contrabass bugle, and Alto Horn
The baritone horn is similar to the euphonium, which was developed in the 1840s, so the baritone came a short time later.
French horn, trumpet, tuba, bar-sax (also part woodwind), baritone, euphonium (like baritone, but with straight top).
The euphonium is also known as the tenor tuba, baryton(Ger.), basse à pistons(Fr.), and sometimes mistakenly referred to as a baritone horn or just a baritone. The baritone horn, however, is a cylindrical bore instrument with a brighter tone (closer to trombone timbre) than a euphonium. The bore of the baritone is often much smaller than that of the euphonium. The euphonium is a conical bore (gets larger is we reach the end) instrument with a warmer and darker tone than a baritone.
Trumpet, French horn, Trombone, Baritone/Euphonium and Tuba.
Euphonium and baritone horn
French Horn Tuba Trumpet Baritone Euphonium Trombone Cornet
Cornet, flugel horn, tenor horn, baritone, euphonium, trombone, tuba.
Euphonium comes to mind. That's a baritone horn.
Euphonium and baritone horn A+
Trumpet, French horn, trombone, tuba, in an orchestra; euphonium and baritone horn in a concert band; tenor horn and alto horn in a British-style brass band.
French Horn, Trumpet, Cornet, Trombone, Tuba, Sousaphone, Baritone and Euphonium.