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New or Different Mechanics in Sonic CD
Spin Dash
Curl up into a ball, charge up in place, launch forward in a spin attack. The Spin Dash in Sonic CD has two differences compared to Sonic 2 and later games: You are not invincible while charging up the dash, and you charge up the move by holding down + jump button instead of pressing jump multiple times. The game uses the same sprites for the charging as for rolling.
Super Peel Out
Charge in place by holding up + jump, release to run forward at full speed. This charges faster than the Spin Dash and you'll have an easier time running up hills, but you'll be vulnerable to whatever you run into. I like the figure 8 style they used for this.
Time Warp
Hit a sign that says "Past" or "Future" and the word will appear next to your life count. If Sonic goes fast enough some sparks will appear around him; maintain that speed for six seconds and you will warp one step to the past or future. Slowing down in the air does not count as being slower for warp purposes -- only the speed when you were launched and when you hit the ground matter, where applicable (there are a few good ways to cheese warps). If you slow down while sparks haved appeared and the word next to your life count is flashing, you will lose the warp completely and will have to hit another sign.
A really short cutscene happens every time you warp, too. It's neat the first time, but gets tedious later. I'm really glad it was cut out of the videos! 
There are two versions of the future: Good or Bad, depending on whether you destroyed Eggman's Machine in the past. The bad future is dark and bleak, and badniks are decrepit/damaged. In the present, around half the badniks are.
Eggman's Machine
This machine spawns badniks, and there is one the first two zones of any stage. The present and future versions are broken down, but the past version works fine. Destroy the machine in the past to release a bunch of flowers, destroy all badniks in the zone and change the future from Bad to Good. Destroy all of these to get the game's good ending.
Projector
In the past variants of the first two zones of any stage, there is a projector featuring the image of Metal Sonic tormeting some animals. Destroy the projector and animals will populate the zone. This has no real gameplay effect except a message at the very end of the game.
Special Stages
Run around a pseudo-mode 7 stage trying to destroy all of Eggman's UFOs to get a Time Stone. You are timed, and hitting water will make you slow down and rapidly deplete your leftover time.
Time Stones
This game's Chaos Emeralds. Collect all seven, get an automatic good ending to the game.