Welcome to my TED talk. Actually, I just want to make this thread because I have the unique opportunity (on a daily basis!) to experience the difference between a variety of instruments that many people seem to have an issue distinguishing. So uhh...let's do something about that. Also, if you'd like, please provide some recommendations for music that I could use for the purpose of comparative demonstration. Needs to be something short and simple because I'm a talentless hack, and don't have huge amounts of free time or super-mega-fiberoptic internet with 200Gbps upload speed.
* Disclaimer *
I will be using German-American type terminology. I am not completely familiar with instrument terminology in Latin-based languages. I am aware that the terminology in French is not compatible with English, so don't bother telling me about it.
So the first thing we need to do is make the distinction between parts, and instruments themselves. A part can be labeled whatever the composer wants to call it, regardless of whether or not that label actually indicates anything. Rather than ramble on about examples, I'm just going to call this context and talk about those contexts. I'll compare the instruments later.
Old timey American Brass Band - 3 part 9'Bb "Saxhorns"
This context comes from the era between the American Civil war and the earlier years of JP Sousa and the popularity of big bands in the US. AFAIK, France had ensembles with similar arrangements.
Bb Tenor = This part is usually the same as the Trombone part, but written in TC. If there are any differences, the Bb Tenor part will be simplified.
Bb Baritone = The soloist/lead part.
Bb Bass = This part is usually the same as the Bass Trombone or 3rd Trombone part, but written in TC. It is entirely possible to find parts requiring a 4th valve. High range rarely exceeds 6th partial G.
British Brass Band
You probably know more about this than I do.
Baritone Horn = Kind of like a lackey instrument for the Euphonium. It does not replace the Trombone.
Euphonium = The soloist/lead part.
(Obsolete) Bb Bass = BBb Tuba part doubled up an octave.
(Honorable mention) Tenor Trombone and Bass Trombone
European Military Band
These sorts of things are not entirely standardized, and these ensembles have actually evolved quite a bit over the years. Compared to the BBB, these ensembles are unique for having the smaller of the 9'Bb pair be the lead part.
Bb Tenorhorn (Baßflügelhorn) = The soloist/lead part. Written in TC.
Baritone = A part that plays below the Tenorhorn. Usually written in BC.
(Honorable mention) Tenor Trombone and Bass Trombone = It should be mentioned that Soviet military bands did not use the Bass Trombone in any form.
Ok. So these are the parts. Now we can talk about the instruments and compare them. I have a ridiculous number of these things, so it's pretty easy for me to do that at home.
I will reference these instruments by number, left to right.
Comparison of instruments
So let's start with the smallest instruments. The part called Bb Tenor in the American context was played by its own instrument which was colloquially known as a "Bb Tenor". My Bb Tenor (2) is not necessarily an average form of this instrument, but it's sufficient for comparison. Historically, this instrument was effectively the same as the British Baritone Horn. The typical differences will be that the American horns have a smaller bore (.470" to .500) and larger bell (~9"+). Due to the demand of a brighter sound, the American instruments also tend to use much smaller mouthpieces. My stock mouthpiece for my Buescher is barely distinguishable from an Alto Horn mouthpiece. My Baritone Horn (1) is a more typical example of what a British Baritone Horn was expected to be in the era just after WWII. Categorically, these instruments are effectively variations of what could be called the "Bb Tenor Saxhorn".
For some reason, the Baritone finds itself in a state of confusion with two different instruments that are both completely different from it. So first I will offer some definition for what a Baritone is and then discuss both situations separately. The Baritone is a very vague, but not impossible instrument to define. In the examples provided, I have three instruments which could be called "Baritones" (6, 7, 8). They have the following shared features. Overall size is distinctly larger than the British Baritone Horn, generally slightly smaller than the American Euphonium, but certainly not limited to that. There is the presence of significant cylindrical tubing. For the ovalform instruments, this will be in the leadpipe and valve block. Piston valve instruments will have a single bore tuning slide located somewhere in the tubing. The amount of cylindrical tubing can differ significantly, but the result is the same. The mouthpiece behavior of these instruments is considerably hateful, possibly bound to a single ratio of throat and cup depth. Categorically, these instruments are effectively variations of what could be called the "Bb Baritone Saxhorn", although the amount of variation can be quite considerable.
The Baritone and Baritone Horn are not the same instrument. They have different sounds, different behaviors, and cannot effectively replace one another. They are both designed to play completely different parts and have coexisted (sort of) in the same ensembles. Using my own instruments as examples, my Conn Baritone (6) is considerably larger than my Baritone Horn (1), being only slightly smaller than my American Euphoniums (9, 10, 11).
Likewise, the Baritone and Euphonium are not the same instrument. Despite what you may have heard or even come to know through colloquialism and social concourse, these instruments were distinguished correctly in Conn catalogs for most of their production. Maybe it was Conn and only Conn, but that's more than nothing. The Euphonium has some rather distinct properties that separate it from any true Baritone. The only cylindrical tubing worth counting on a Euphonium will be the valve block (~5%). As a result of its design, the Euphonium is able to use a wide variety of mouthpieces without issues in intonation or tone development. Despite whatever is technically possible, if an instrument has a tuning slide in the leadpipe, it is not necessarily a Baritone, but it is definitely not a Euphonium.
The instrument playing the Bb Bass part, typically known as "Bb Bass Saxhorn" for Americans, doesn't seem to have enjoyed much popularity outside of France. My Bb Bass Saxhorn (12) provides a clear example of what the French instruments were like before WWII. There is a short leadpipe with small shank reciever, dual-bore tuning slide, some additional length of cylindrical tubing before the valve block, and a relatively large bore size (~.590") which explodes from there. The overall size is smaller than you might expect being roughly equivalent to the American Euphonium (9, 10, 11). Once again, unlike the Euphonium, mouthpiece behavior is hateful. More importantly, these instruments are designed to play better in the low range than the smaller horns. I'm not able to determine how long these instruments found usage in English speaking environments, and I haven't been able to find any American produced instruments that match the French instruments. I do know that these horns were considered roughly equivalent to Euphoniums in BBBs, and used to broaden the sound of the bass parts until further standardization disallowed this.
Finally, getting back to the Euphonium, there's a lot more to say about it than just how good it is. Actually, it's interesting that the Euphonium failed to make itself a staple of the orchestra given that it was a complete world beater in everything else. The American Euphonium (9, 10, 11) really should get addressed first because it is an underrated instrument. Despite being often called a Baritone, it is a real Euphonium having the correct amount of cylindrical tubing and proper behaviors. Concerning the original context of early American bands, the Euphonium physically replaced the instruments that played both the Bb Baritone and Bb Bass parts. The Euphonium continued to play the Bb Baritone part, while the Bb Tenor and Bb Bass parts went to their respective Trombones. Having lost its smaller lackey instrument, and having had to be good enough to cover two different roles, the American Euphonium is specifically designed to be extremely flexible with a powerful, focused tone. In comparison, the British Euphonium (13) is not as much larger as its reputation suggests. Rather, most of the difference comes from the bore sizes with the British instruments being much larger (.580" vs .560"). This difference in bore size, as well as other subtle differences, prevent the two instruments from having matching tone qualities.
Finally, there is the matter of the various Bb Tenorhorns. The smaller sorts of instruments, like my Bb Altohorn (3), may have been referred to as a "Bb Tenorhorn", but they were not used to play Bb Tenorhorn parts. Rather, this part was specifically played by the type of Bb Tenorhorn also known as "Baßflügelhorn". This instrument is larger than the British Baritone Horn, and not acoustically compatible with it. While it's not possible to pinpoint the exact size for any type of instrument, you can expect a Bb Tenorhorn of any type to be smaller than the Baritones that it is expected to pair with. These can be found in Tubaform, Ovalform (4), and Kinderform (5) designs, with subtle differences to be expected. In general, these instruments should have the capability to perform as a lead instrument despite their smaller stature compared to the other tenor lead instruments. The typical mouthpiece will be smaller than even vintage Euphonium mouthpieces. This is the instrument playing those "soaring" parts that are typical of older European military band music.