As already noted by the OP, the question is indeed very broad, and this answer obviously takes a very subjective approach to what is being asked, so I'm mostly going to address music about the cited composers (Williams, Horner) and those that could be considered affine to their "classical" style.
Conceptually speaking, they do not differ a lot.
There are obvious differences, and what you point out as a "different sound" between Horner and Wagner, is the difference in "sound" of each composer.
For instance, despite belonging to the same era, using a very specific musical style and almost always identical instrumentation, you can usually recognize if a composition is written by Bach or Händel even if you have never listened to it.
"Classical" film music also tends to use much more distinct characters for any given piece (except for longer "suites"), and that's because they usually have a single composition for a very specific scene, which has to portray what the director wants to highlight. While they do have "developments" of themes, their forms rarely use classical layout forms, as what happens on the screen is what actually creates those layouts, sometimes leading to section repetition, abrupt changes in character, etc. Another important aspect is that some scenes require some time for their development, forcing the composer to extend sections that are sometimes not really "musically important", and that's because the music shouldn't distract the viewers from what's happening, possibly leading them to beliving that something is happening while it's not (yet).
Orchestral film music is also often "stereotyped", with very large string sections for moody and melodic passages, powerful brass groups for heroic and thrilling scenarios and, most importantly, a much larger percussion palette (remember that most instruments now commonly used in modern orchestral compositions were rarely considered for classical compositions, and some were not even invented yet).
Note that many composers that could still be considered "classical" in a broader sense, have a very different approach, sometimes radically. For instance, Jerry Goldsmith is usually closer to the mentioned composers, and a bit more than Alan Silvestri could be, but still somehow more modern (including a larger usage of electronic instruments or a sound that resembles them). And while Hans Zimmer adopted a more symphonical approach in his maturity, his style is clearly very different. Other composers chose to follow an even more different attitude, while mostly keeping a typical orchestral sound, like Elliot Goldenthal. It's also worth noticing that sometimes even normally-classical composers take a different path that is closer to their formation: Williams is known to write interesting jazzy music (see the score for Catch Me If You Can, or "Banning Back Home" from Hook) as much as Danny Elfman often goes back to his pop/rock roots.
That said, consider that those "classical" composers also explicitly borrow from previous musical styles (sometimes for personal choice, others, obviously, due to film requirements) which is something that happened much less often in the past, and while doing that they also "modernize" those styles with instrumentation and orchestration methods that are not typical of those styles.
Take for instance the suite "The Forest Battle" from Star Wars: Episode VI - Return of the Jedi:
You can clearly hear elements from Prokofiev and Shostakovich, but with orchestration choices and instrumentation that were not exactly from those eras (even considering their rather modern approaches). Lots of winds, fast and bright themes, but also strings used much more as a sound palette, a more "romantic" approach for more broad/heroic themes and, obviously, a lot more of colorful percussions.
It's also worth noticing an interesting aspect about Horner, since you mentioned him: he has been known (even in his life) as a nasty borrower, sometimes taking exact excerpts of other earlier composers and with very little adaptation.
Take for instance "Death of Tybalt" from Prokofiev's Romeo and Juliet (note the section starting at 1:18):
Then listen to the soundtrack version of "Stealing the Enterprise" by James Horner, for the score of Star Trek III:
The initial theme of the violins is not only the exact copy, but it's also in the same key. The orchestration is a bit different though, including harp arpeggios and a glockenspiel (or crotales), which make it a bit lighter in tone, considering that the scene has some elements of comedy.
The violin part at the beginning didn't survive the film score, some say that the producers asked him for less "movement" at the beginning in order to provide a better effect in the development of the scene, but considering that in some cases his "borrowings" also led to litigation (see the link above), it's also possible that they required him to remove it for copyright reasons: the film was made in 1984, but Prokofiev died in 1953, so Romeo and Juliet wasn't public domain yet.
 with "classical" I mean music that uses a symphonic orchestra with compositions that stylistically resemble late 19th-early 20th century music; also consider composers of the Hollywood's Golden Age, like Max Steiner, Alfred Newman and Erich Korngold (the latter of which is known as the most influential composer in "modern" film music, and also a known inspiration for John Williams himself).
 there are some rare exceptions, and the most known is that from the final chase scene of E.T. the Extra-terrestrial, as Spielberg loved so much the music Williams wrote that he actually re-edited the scene to match the whole suite in order to leave it almost unchanged.