I have both, so I'll kick in my two cents worth.
They're completely different. Completely different feel, completely different grip, and completely different heft.
IMHO, it depends on what you want to do with it. To me, the JAE is the stock I'd have for hunting or if I spent long hours on glass, or shot mostly from the prone or fixed position. It is just so comfortable in the prone (you could fall asleep on it easily), or off a tripod. The fit and finish is top of the line. If you're OCD about stuff, the JAE is your deal. Mine is a very early version, so it is much heavier than the newer versions, and thus I can't really comment about weight (though mine is fairly heavy). I know they shaved some weight off of it in the current version.
We're I to focus on just a comp PRS/NRL gun as it's intended use, than I think the MPA would be my choice. Very light, vertical grip with thumb shelf, integral ACRA dovetail, and tons of places to hang stuff from. A very nimble stock when having to transition between props and shooting windows.
Bottom line: You need to get behind each one to decide if the extra cost of the JAE is worth it to you. I kind of have to shake my head at what JAE stocks go for today (though worth every penny I'm sure). But then again, I put down a deposit back when it was just a prototype, had it's own proprietary magazine, and waited almost four years. It's one of the stocks I know I'll keep until it gets passed down, after I die. The MPA on the other hand, I consider a tool, and have no issues with beating the shit out of it.