The difference may be negligible but before I load up on ammo, thought I would ask if one of these rounds is silenced better than the other using a 7.62 can.

Also, if a 7.62 can reduced a 308 by 30 decibels, what should I expect from a 223, more or less reduction?

The same caliber bullet pushed out at the same velocity from the same length barrel is going to sound the same, regardless of whether the case is a .223 case or a 5.56 case.  Different bullet weights, velocities, and chamber pressures might change things a little bit, but unless you go to an extreme, you probably won’t really notice it.

You’ll probably get a slight (1-2 dB) reduction using a 5.56 sized can instead of a .30 caliber can when shooting 5.56/.223, just due to the closer fit of the baffle holes to the projectile diameter.  Good 5.56 cans can get a sound reduction of 33-35 dB from unsuppressed.

Technically .223 should be ever so slightly quieter than 5.56 because of the slightly lower pressure but that's not something that you'll be able to discern with the human ear. You typically aren't giving up much performance with a .30 cal can vs a 5.56, usually only about 2db difference. The upside is that at the shooters ear the performance should be the same or slightly better due to the larger .30 aperture causing a little less back pressure.

Ok, makes sense. I was thinking the larger, more powerful 7.62 can would have more effectiveness on a 223 but understand your point.

Well in my non scientific tests .223 has less blast by a noticeable margin.  
M855 vs PMC 223 out of a 10.5 suppressed AR.

Supersonic rounds will be loud regardless of which can you use.