Brass thickness varies. As a general rule, yes, 5.56-specific brass (such as LC) is going to be thicker (and thus heavier) than .223 brass. But not always. The weight spread overlap is so great, that there's not much point in worrying about it.

If you are loading "brass is brass" empties, for practice and plinking, then load to .223 loading data. You are unlikely to notice any POI or accuracy difference, between .223 and 5.56 load specs, using the same bullet. We all know it is an American birthright to be able to buy a sub-MOA AR right over the counter, but most aren't. And most shooters aren't, either.

If you have to have 5.56-performance ammo that you load, then use a single batch of brass (LC is a good source) and load it up for your rifle.