Why difference in surround speakers?

It is difficult to say much since I cannot get your link to work but there is generally no difference at all. However, some trapezoid-shaped speakers have multiple drivers arranged in bipolar or dipolar configurations in order to create a more diffuse radiation. Some recommend and prefer this to a regular monopole speaker but I do not.

A direct radiating speaker does just that, radiates the sound pretty much directly at you. A design say like the Axiom Qs8's which have 4 drivers firing in phase up/down/left/right envelope you in the experience which surrounds you as the recording engineers intended.

Clearly, that depends on the particular engineers and may apply more to HT than to music.

Yes, most people would use di, bi, or quadpolar speakers for HT and direct radiating for music. Interesting my Axiom Qs8's do great for both applications.

The recording engineers mix in a studio with 5 direct radiating monopoles. Any ambiance necessary is mixed into the soundtrack by the engineer and does not rely upon the speakers to be reproduced properly. Ever since the advent of truly discreet 5.1 soundtracks, diffusing speakers became a thing of the past, although they were at one time preferred with matrixed surround sound such as DPLII. But old habits die hard.

There is no clear answer for this unfortunately. As you've already seen here, some prefer one or the other. It depends if you want the sound the way it is in the mixing room or more like in the theatre where there are many surround speakers creating a fuller sound. I prefer the latter, because after all, movies are meant to be watched in theatres and not mixing rooms.

The direct radiating speakers create a LESS full sound, while the others create a more theatre-like sound.

I disagree with that statement. Monopoles create a very full sound but, due to their specific and defined radiation pattern, are more critical to set up and will suffer if placed inappropriately. Dipole/bipole surrounds bounce their radiations off the room surfaces and the sound arrives at the listening position from many directions, resulting in the impression of a diffuse source. This results in somewhat less specificity of source direction but a much greater tolerance for less than critical placement. If you look at the many threads here on setup and speaker placement, it will be apparent that many, if not most, posters cannot place all their speakers exactly where they should be and, therefore, benefit greatly from using bipole/dipole surrounds. In fact, I see no down-side to it for HT, at all. (Discrete multichannel music, of course, is another story.)

BTW, all the speakers you are referring to are 'direct radiators. The only common non-direct radiators are those with horn-loading or quarter-wave pipes.