Christmas Rushed: The game in its completed state was too ambitious for it to be released as a whole in such a short time. As well, McDonald's ended up pushing Happy Meal toys before the game was released, forcing Sega's hand in the entire thing. Sonic 3 was released in February 1994 and & Knuckles would be released eight months later.
Copy Protection: Pirate copies of the cartridge obviously do not have the lock-on slot. This means a huge chunk of content will be unavailable, so even though the game is still fully playable, it's a lot less interesting.
Fan Remake: Sonic 3 Complete is a fairly well-known hack that polishes the game in areas that the official Sonic 3 & Knuckles didn't (fixing glitches, giving Super Sonic actual sprites for certain things) as well as allowing a plethora of options such as choosing individual tracks between Sonic 3 or Sonic & Knuckles and allowing the ability to play Sonic 3 or Sonic & Knuckles standalone in the same game. It also added some goodies like restoring the "Beta" level order (Flying Battery after Carnival Night), and the option to do the Super Peel Out from Sonic CD. For more extras, you can even create IPS patches that change the game even further, such as changing Sonic back to his Sonic 1 and Sonic 2 sprites, with new sprites to accommodate them for all of the actions he could only do in Sonic 3 & Knuckles.
Manual Misprint: The two Badniks from Flying Battery Zone in Sonic & Knuckles first showed up in the Sonic the Hedgehog 3 instruction manual.
Screwed by the Lawyers: According to Cirocco Jones (mistakenly credited as Scirocco in the Sonic 3 credits), there's a huge legal mess concerning the music used in the games, likely due to unpaid royalties, improperly crediting the musicians, or (as was the case with Dreams Come True), licensing issues. Considering that two of the music composers (Michael Jackson and Brad Buxer) directly sampled songs from their respective discographies, the latter is the most likely.
It's generally believed this is the reason why a Taxman/Stealth remaster like the first two Genesis games is unlikely, as Sega has only been able to release direct emulations of the original game or recoded versions with different music (as seen in the PC version).
There tends to be confusion as to why the original Sonic the Hedgehog could not be locked in with Sonic & Knuckles. The two most popular explanations are Knuckles would have dramatically altered the color pallet or that Knuckles gliding on conveyor belts in Scrap Brain Zone would cause the game to crash. Both of which were solved by the famous ROM hack Knuckles the Echidna in Sonic the Hedgehog as well as any future ROM hacks starring characters other than Sonic.
Despite what your third grade friends (and basic logic) told you, it's not possible to beat Fire Breath the first time you see it and stop it from burning Angel Island. No matter how fast you hit it, it'll successfully set off the firebombs and escape with its robotic life.
What Could Have Been: Flying Battery Zone was originally planned to be in the Sonic 3 half, between Carnival Night and IceCap (and Sonic would have improvised the broken door as a snowboard). Time constraints intervened, and it (along with at least "Mushroom Valley" and Sandopolis) had to be cut and moved to the Sonic & Knuckles half. In the end, this was probably a good call, as it helps to balance the emerald collection in the combined game.
Additionally, the Super Emeralds and their powers were not part of the original plan - according to Takashi Iizuka at Sonic Boom 2013, they were only added after the game was split so the player would have an incentive to go through another set of Special Stages.