It would really depend on the particular variant and manufacture.
Generally speaking, FAL's aren't the pinnacle of MBR accuracy, but neither are M1A's. Somewhere along the line, M1A's and M14's obtained this legend of mythical accuracy, which just isn't true. FWIW, the "threshold" of acceptible M14 accuracy was around 5 inches, according to the "real" SA and the Ordinance Department. However, most seemed to range from 3-4.5 MOA with M80 ball.
While M14's can shoot very accurately, M0A or less, depending on ammo, models suitible for defensive use will be in the 3 MOA +/- range, again depending on ammo.
I hate this term, but a "milspec" FAL should shoot 4 MOA or less with standard FMJ ammo. However, civilian FAL's can have quite a bit of variance. It also depends on the model/origin of the FAL. IE: 50.63, Congo, Israeli, FALO, etc....Certain features like an HBAR or improved sights can aide in accuracy. In any event, a competently built FAL composed of good parts should be 4 MOA or less with military FMJ ammo. An FAL made by DSA could be much less, however price will be much higher.
Remember, that the FAL is similar in concept to the AK. Reliability is more apparent than accuracy. Its a battle rifle, not a target gun. I view FAL accuracy like I view AK accuracy, academicly. So long as my FAL shoots the specified 4 MOA with ball ammo or if my AK is "minue of dishwasher" I am okay. Buying an FAL for accuracy is like taking a stripper to breakfast and using your real name, its just misguided and all wrong.
AR's have really spoiled us, as has good quality ammo. AR accuracy isn't the rule for infantry rifles, its very much the exception. Generally, you aren't going to get AR15 accuracy from an MBR without some $$$.
That being said, both the FAL and M14 aren't the most intrinsicly accurate weapons ever devised. However, both have been around a long time and accuracy can be improved. You need to establish how much accuracy is necessary for your mission. Most of the "rack grade" MBR's will be in the 4 M0A or less range, which is plenty for a defensive rifle. If you don't concur, consider the fact that combat distance shrinks every conflict. Both the IDF and DoD have concluded that MOUT engagements take place at around 110 yards, +/-. Civilians engagements will be considerably less....Also consider the dynamics of engaging a moving human adversary at say 300 yards with ball ammo and iron sights. Not very promising, even for a superior shot....
As a DMR, more accuracy will be required, as will another platform. An AR10 is the best choice for a SASR, followed distantly by a G3.
Comparing a "standard" M1A to a "standard" DSA STG.58, I believe the M1A would have a slight edge in accuracy over the FAL. Perhaps around .25 to .50 MOA... Spending more money will get you more accuracy, though. DSA makes some exceedingly accurate FAL's, some delving near the MOA mark or below. Obviously, SA makes M1A's that will accomplish this too. The difference is that an accurized FAL will still be a suitible infantry weapon, while an MOA M14 will not be. This is why the M21 and M25 were abandonned. Ultra accurate M14's are too hard to sustain and lack the reliability of thier "rack grade" cousins. This isn't as true, or true at all, of accurized G3's and FAL's.
IMO, MBR's really come down to personal preference and overall sillyness. As combat weapons, they are nearly as antiquated as a 98K or P.14. They also have similar relative performance in terms of accuracy and reliability. Each has thier own benefits and drawbacks. I would advise you to figure out what your "mission requirements" are and then find a rifle that fits within that umbrella. If you are just looking for a rack-grade MBR, pick one that you like.
As to which one "shoots better", thats entirely personal preference. Personally, I find the M14 to be a much more comfortable "target" gun, due to its excellent sights and comfortable stock.