I had the electrician install a dedicated 30amp 220v circuit to my garage when my house was built for future use. He installed a typical 4 prong dryer outlet. I have an old table saw the runs a 20amp 220v motor. I'd like to change out the outlet so I can use it.

Here's the questions. The line is 10/3 wire. The motor runs 12/2 wire. To change the outlet, I need black,white,ground. What do I do with the red on the line from the breaker box?

If the saw is 240 volts and no neutral, cap the neutral, connect the red, black, and ground to the receptacle, then at the panel, connect the red and black to a 2 pole 20 amp breaker.
The white is likely what you don't need.

The saw can actually be run on either 120 or 240 according to the name plate. It has just 3 wires, which includes a green. So if the white (neutral) is not needed in this case, why does code require a 4 wire (3 w/ground) cable for a 240 circuit

The requirement is based on the need to separate Neutral and ground.

Neutral and ground circuit design serve two distinct purposes.

Under current code, the only place that Neutral can be joined to Ground is at the main electrical panel board.

Neutral is a current carrying conductor that is grounded at the service.

Ground is a "0" reference point that will carry an objectionable current at a high enough speed to trip the breaker and cut the circuit off.

I'm paraphrasing to try to make it easier for you to understand, I hope I helped.