I just installed an appliance that called for a 240v 20amp service. I ran the line and attached a double pole 20 amp breaker (It had two separate throws). The appliance would not work. I installed another 20 amp breaker with a common trip and it runs fine.

What is the difference?

I have a Cutler-Hammer BR series box. Both breakers where 20amp double pole breaker for the BR series. What's the deal?

The first one you tried was NOT a double-pole breaker. It was a twin, or tandem, breaker.
A twin breaker connects to the same single buss in the panel so you were only getting 120v to the appliance.