They can be heavier. I have some Starline +P brass that is a little thicker. From Starline's site:
"The {Starline} 45 Auto+P is a strengthened version of the 45 Auto with the same external dimensions. A thicker web and heavier sidewall at base strengthens the case in potentially unsupported areas. This case has approximately 2 grains less internal water capacity than the standard 45 Auto. "
Two grains of water capacity in the .45 ACP shooting Bullseye is equivalent to about a third of a grain less powder to reach the same peak pressure. Since the specific gravity of cartridge brass is 8.53, such cases weigh about 17 grains more than standard cases, if you want a way to check.
I don't recall anyone recommending running a magnet over the brass yet. I do that because some of the varnished steel cases I pick up resemble the patina on brass that's been on the ground awhile, so I sometimes miss culling them on visual inspection. Steel cases are hard on your dies.
Sevens is correct on accuracy. Years ago National Match shooters observed military issue match ammo was more accurate than commercial. It turned out the pitch seal in the military cases improved start pressure which improved ignition consistency. I don't intend to start putting pitch in my brass (be useless with lubricated lead bullets anyway), but this does tell you consistent grip on the bullet contributes measurably to accuracy in a gun that's accurate enough to let you see it. In a gun that's not match accurate, I don't expect to be able to see the difference, so this is a judgement call.
I always use brass that I purchased originally new in bulk for match shooting, and have been careful to put through the same number of load cycles, so work hardening is evened out among the lot. For my school gun and shooting steel plates and silhouettes and running dueling trees and whatnot, I just load whatever I pick up and don't worry about it.
I used to sort out and toss or leave behind any R-P brass ran into. The mouths are so thin it doesn't size down much and would often work harden to the point it wouldn't resize enough in my old Lyman carbide die to hold onto a bullet at all. In my Dillon carbide die (narrower) it seems to work OK, and the same might be true of other current production dies. You'll have to try this for yourself.