Miscellany

Japanese to English (NES)

The English version naturally has the text translated, and tweaks the sign.

Birdo's words of encouragement are translated directly in the English version.

Wario's dialogue is translated, though some of it appears to be shortened in the English release.

The name entry screen is shrunk in the English version due to having a smaller alphabet.

License was renamed Rankings in the English release.

This one can't really be shown in images, but the input for placing an object beneath Toad is far clunkier in the Japanese version.
You have to press the A and B Buttons simultaneously to do this simple task, but due to some really finicky timing more often than not you just drop the object in front of you.
The English releases simplify this significantly by pressing Up on the D-Pad to drop it beneath you.

American to European (NES)

The European version runs slower than the other versions, due to being reconfigured for 50Hz display. The only noticeable quirk with this is that the coins drop insanely fast at the end of a Round Game match, though everything else transfers fine.

Strangely, the European version features brand-new music replacing most of the soundtrack! Sadly, an NSF of this version has not yet been dumped so I can't make a totally comprehensive comparison, but I can verify that the following have not been changed.

• All victory jingles (except Vs. Mode's results screen)
• Stage clear
• Game Over
• Time Race
• VS.
• talking to Wario

There are some curious additions. In the NTSC releases, the Rankings/License screen reuses the main menu music, but is given a unique tune in the PAL version.
The 1-Up sound after collecting thirty coins has also been changed.

Unused materials (SNES)

When you beat a character in Vs. CPU, they get turned into their head (or some kind of head-related representative). An extra frame of animation is visible in the tiles via vSNES, but isn't seen in-game.
The defeated character flickers briefly after being defeated - if one is in the mood for speculation, it might be that's when the extra frame is played, but the flicker kind of renders it invisible. Either way, it's incredibly minor and pointless, so it's no big deal.