You can drift in both, it just takes a certain amount of throttle control, but if you want an easier and, imo, more fun drift experience, play NFS Underground 2.
The drift events will take some getting used to, but once you figure it out you'll easily get 100,000 points above the 2nd place guy.
And drifting in the open world only really takes a tap of the e-brake and you'll be going through every corner sideways after a few expected foul ups
The only downside is you'll be missing out on all of the exotics from MW and on
EDIT: After watching the video, just stick with it and find the right amount of throttle for the car you're using. But if you find yourself getting tired of trying, give Underground 2 a shot, the cars are much easier to control than in all of the NFS's after it.