I'm installing a whirlpool tub in my master bedroom. The elctrical requirement is a dedicated circuit for the pump and a circuit for the heater. Both will need to be GFCI protected. I ran a 12/3 cable thinking I can use a double pole 20 A GFCI breaker. However, I have heard it will not work as it cannot monitor both circuit which I don't understand why not since it's double pole. I have not purchased the breaker since I don't want to spend the $$$ on something that will not work. Thank you for your feedback.

You haven't specified the electrical requirements: e.g.

Heater 15.4 A 240 V 20 A GFCI Circuit breaker recommended
Pump 1/2 HP 120 V 15 A GFCI Circuit breaker recommended

You should also specify the distance to the breaker box.

The electrical requirements are:

1) heater - 120 Volts 12.5 amps
2) pump - 120 Volts 7.5 amps

The distance from the breaker box to the outlet is 60 feet.

So you are saying you ran just one 12-3 cable up to the tub, right?

Yes, I only ran one 12-3 cable up to the tub.

Then a 2 pole GFI breaker will work fine.

Actually if you ran a 12-3 with ground, your probably OK with a 240 V 2 pole 20 Amp tandem GFCI breaker if the distance is 59 feet, 3% drop at 59 feet, and the heater load is classified as operating more than 3 hours at a time. The pump load isn't a problem.

The neutral will only carry the difference current of 12.5-7.5 amps, so you won't overload the neutral. Both circuits will trip simultaneously which would be an advantage.