The biggest difference between the two is that the JL amp is regulated. In a normal application you're going to be able to squeeze 1000 watts out of the JL amp even when the voltage in your car drops, whereas the Alpine is going to require a steady 14.4 volts to maintain a similar output.
How big of a difference will it be? Well, I used to own an older model Alpine V12 amp running on a stock 110 amp alternator, and when I upgraded to a high output alternator the difference was actually shocking. My bass hit much harder and played cleaner than ever before.
Both amps are nice, but for the average user if you have the money the JL will probably be the best sounding choice for the sake of simplicity.