Sonic 3 and Sonic and Knuckles were originally going to be released as one game, but it ended up being cut into two games. Sonic 3, when compared to Sonic 1 and 2, is a MAJOR step. With more options for verses mode and saving ability, Sonic 3 introduced a lot more options for Sonic gaming fun.
The game also introduces the character Knuckles the Echidna, whom of which is the guardian of Angel Island, where Dr. Robotnik (Eggman) lands after his defeat from Sonic 2. Dr. Robotnik manages to convince Knuckles that Sonic and Tails are the ones who want to cause trouble, so Knuckles often gets in the way and makes things difficult for Sonic and Tails during the entire game. In verses mode, you can play as Knuckles as well as Sonic or Tails.
In the usual game, you can play as Sonic and Tails, Sonic alone, or Tails alone. One thing I really like about Sonic 3 as opposed to Sonic 2 is that Tails can fly, which often gives him a great advantage over playing Sonic. When you do 2 player with Sonic and Tails, Tails can often fly Sonic to high places, but it's not a long time before Tails gets tired.
Levels: The levels are big and often challenging, usually taking a long time to complete. However, one of the reasons why Sonic 3 is flawed in my opinion is that there are very few levels (6, including the final boss which is not a separate level from the last), which is a small number compared to it's predecessors (Sonic 1 had 6 levels not including the final zone, but keep in mind that the levels in Sonic 1 have 3 zones, not just two; Sonic 2 had 11 levels).
Best level: Ice Cap Zone (Level 5)
Graphics: Beautiful and clear, MUCH improved over Sonic 2.
Bonus/Emerald Rounds: The idea of a bonus round is introduced with Sonic 2. With the new bonus round (the gumball round) you have a chance to get more rings/lives/shields. It's often a quick escape from a potentially bad situation. The emerald rounds are also new; basically consisting of Sonic or Tails having to gather up blue spheres and not touch the red spheres in order to get the emeralds. I find that those emerald rounds are generally easier than the ones in Sonic 1 or 2, which I am not a big fan of at all.
Characters: Sonic and Tails, Sonic, and Tails. Knuckles as well in the verses mode. Not too bad a line-up, I believe.
Music: I don't really like the music in Sonic 3. Most of it I believe lacks the catchiness of the music from Sonic 2 or Sonic and Knuckles. I believe that Hydrocity Zone and the Ice Cap Zone themes have their charms, but nothing really stands out as being all that good. Thankfully, the music was improved a LOT in Sonic and Knuckles. However, an interesting thing that is new with Sonic 3 is that the themes for the different levels change slightly for the different zones of the level; for instance, the theme for Hydrocity Act 1 is SLIGHTLY different then the theme for Hydrocity Act 2. Often, you have to really listen to the themes to hear the differences, but they are there. This trend adheres to Sonic and Knuckles as well.
Bosses: I find the bosses for Sonic 3 to be extremely difficult at times. They are much more difficult than the bosses in Sonic 2, which can often be beaten in a matter of seconds. The bosses in Sonic 3 take a lot of patience. Even some of the bosses in Sonic and Knuckles I find are often easier than the ones in Sonic 3. Sometimes, the bosses can be so difficult it can take a matter of rings (and even lives) to beat them. However, the final boss I find is actually a little easier than expected. Also, for the first time in Genesis gaming, you are able to have rings to defeat the final boss.
Basically, for Sonic 3, the thing that was improved the most over Sonic 2 was special features (and graphics). Not only did they add a verses round, but they added more options such as time track (try to beat your best time w/ separate characters!). Also, the saving mode allows you to save where you left off from, and also saves all your lives and continues. Once you've completed the game, you can go back and play any level you wish on your saved game. You may delete any saved game that you want at any time, and you can even play no save! Tails can fly, which is an awesome feature. And, another added bonus is the addition of shields to the game. In Sonic 2, the only shield that existed was the protection shield, good for one hit so Sonic could be spared his rings (or a life). However, with Sonic 3, three new shields replace the old protection shield. The flame shield protects Sonic from any flame or fire-induced harm (even lava!), the water shield protects Sonic from drowning, and is VERY useful in Hydrocity (and other levels, since Sonic 3 has a LOT of water), and the lightning shield, which protects Sonic from a lot of electric hazards and also attracts rings by magnet. Sonic can do specialized moves with the shields, but even when lacking a shield, Sonic can do an insta-shield, which is useful when a hazard is near. Tails can also use the shields, but he cannot do specialized moves with them.
Overall, Sonic 3 is a great game, but does lack in some departments. If you want a more intense gaming experience, buy Sonic 3 and Sonic and Knuckles together and play Sonic 3 and Knuckles. Sonic 3 and Knuckles is quite possibly the best Sonic game ever, and eliminates all the flaws of Sonic 3 and Sonic and Knuckles.