For instance, would a glitch I've seen in speedruns still work? Or have the roms been at least a little bit patched before they were packaged on the SNES Classic?
With the exception of Star Fox, it is my understanding that these ROMs are the same data as those on the original games.
That said, these games are running on a Nintendo emulator vs. actual processing hardware so behavior will be different.
For instance, a friend of mine on another forum posted this video of a glitch in Donkey Kong Country:
From the video, you can see that the vine in DKC keeps swinging in the background process, even though the image is paused on screen. This might be a minor detail, but it highlights the fact that even a rock-solid, 1st party emulator isn't perfect.
Now, regarding Star Fox. I imagine the gameplay of Star Fox should be 100% the same, however they did encode one small change. In order to play Star Fox 2, you have to beat the first level of Star Fox 1. Once you beat the first level, the game is interrupted, a package appears, and shows Star Fox 2's game icon being placed in the menu. Obviously, for this interaction to have occured, Nintendo would have encoded some type of trigger or hook into Star Fox 1.
Of course, there is the chance that instead of modding the SF1 ROM, Nintendo could have found a crafty way to monitor the state of the game from within the emulator and only when the first level was complete would they flash the SF2 unboxing graphic. In which case, SF1 would 100% be the same game, along with all of the others.
Ok, I completely forgot to even consider Star Fox 2. For those who are unaware, prototype carts of Star Fox 2 have been found, the ROM has been dumped and leaked and people have been selling this game hacked into SNES carts that already have FX chips.
Again, I yet own a SNES Classic so I can't tell from experience, but from a little research, Nintendo has made some minor graphical changes to the Star Fox 2 logo, as well as a few minor in-game details. They've integrated a few additional settings (for instance, the option to choose Mono/Stereo output) and updated the copyright details within the game, which makes sense.
However, it's hard to consider this a change to the "original" since most would consider original content the actual version of the game released to the public. Regardless, there are changes in the game that differ from the known final/near-final builds of Star Fox 2 that have long been available on the internet.