One word: immobiliser.
Your car can't talk to the new keyfob. When you don't have the keyfob the computer is set up for, the car will start then stall. That's how the built in immobiliser works, it likes to get a car thief's hopes up before shattering them and causing them to angrily steal a stereo they won't be able to use without the code instead. Or set fire to it, depending on how much of an anger management issue they have. You need the old keyfob.
Fun fact: For a while I was having trouble with my ignition where if I removed the key it would immediately lock up and it would take 5 minutes of f*cking around to get it to turn again. I got around this until I got a new ignition by putting my key in a different keyfob (because those keys will paralyse your hand if you try and turn them without anything on the end of them) and leaving it in the ignition all the time, even when I parked at the supermarket or whatever and I would take the keyless keyfob that actually belonged to my car with me and use it to lock and unlock the car. But, having the wrong keyfob on the ignition key meant my car (a VX Berlina) would start then stall, just like you're experiencing! So I had to get in the car and "unlock" the doors again holding the keyfob near the ignition and pressing the button and then start the car (you get about 30 seconds before the car forgets and you have to press it again). Once I did that it was fine, I could chuck the real keyfob anywhere and the car would stay running (I used to just chuck it in that little pocket under the stereo. Cool story.). Eventually I got a new ignition barrel and put the keyfob that belonged to my car on the new key and I've been rocking that setup without issue ever since (about 6 months or so).
tl;dr - Immobiliser. Your car wants the original keyfob signal or it thinks you're stealing it. The new one can't talk to the ignition. Either reprogram your ECU (more trouble than it's worth) or put the old keyfob on the new ignition key. Problem solved.
