This is really for the newer people here, but I thought I would take it upon myself to explain what is meant by the different schools of playing. I was trained primarily in the French tradition, so if you are a Rascher or Teal person, please feel free to add your comments. I'm not trying to cause debate or contention, but since people have been asking, I thought I'd expalin it a bit better. Let's not talk about why there are feuds or anything else, please.
French School- Can be traced back to Marcel Mule. Prominent players include Frederick Hemke, Eugene Rousseau, Daniel Deffayet, Jean-Marie Londeix and Claude Delangle. Equipment is exclusively modern. (And normally Selmer or Yamaha) Mouthpieces ususally have "shaped" chambers, such as the S-80, Rousseau NC, or Vandoren. Methods include linear etudes, some scale work, articulation work, and repertoire. Tone tends to be brighter and articulations tend to be lighter than the other two schools. There is quite a bit of crossover in training with the American school.
Rascher School-Can be traced back, obviously, to Sigurd Rascher. Prominent players include Carina Rascher, David Bilger, and Paul Cohen. Equipment is usually vintage, especially Buescher. (Many Rascher people feel that modern horns have gone away from Adolphe Sax's original design). Mouthpieces are exclusively round chambered and resistant, such as Caravan and Buescher. The most important feature that differentiates Rascher training is the use of overtone exercises. Tone tends to be darker, articulations heavier, and vibrato is slower than the other two schools. (although this varies.) There is very little crossover training with the other two schools.
American school-Can be traced back to Larry Teal, although there are very few "pure" American school players. Prominent players include Donald Sinta, Michael Hester (who has had a lot of French training also), and Steven Mauk. Equipment is usually modern (Mark VI or newer). Mouthpieces vary, although many American school players use Selmer LTs for obvious reasons. Tone is somewhere between the German and French schools, and the vibrato is usually terminal in imitation of flute or voice. Most American school players will have French training as well.
There is one other school of playing, which I hesitate to mention, but in my studies of this subject I think it's worth explaining. I will NOT give anyone specific names on this:
"Lunatic Fringe"-This is my name for teachers who have created cults of personality around themselves and do not acknowledge any other influence. Equipment is usually EXACTLY what said teacher uses and all tone, etc. is in imitation of that teacher. I will leave you to your own conclusions on this one.