The AR is hands down more accurate, and the FAL is hands down more rugged and reliable.
As far as accuracy goes, the AR-10s are usually around 0.75-1.25 MOA with decent ammo. The FAL's vary between 2.5-4 MOA or so. The reliability difference is harder to quantify. I can't really put a number on it like with accuracy, but I can tell you that the FAL has a reputation as one of the most rugged and reliable firearms ever made. It is up there with the AK in that regard. The AR-10, while it has one of the best reputations for reliability among the various makes of .308 ARs, doesn't quite have the same reputation for reliability as even its smaller relative, the AR-15.
I would say the FAL would make a better battle rifle, while the Armalite would make a better range rifle.
If you want another option that has almost the accuracy of the AR-10 with almost the reliability of the FAL, you might look at the M-14 as well.