Both have the same bullet diameter of 7.62 milimeters. The 7.62 is the same diameter as used in .30:06, .308, .30 cal.
You are partially correct they all do share the 7.62 designation but that only pertains to the bore dia. or measurement across the lands. 7.62 mm = .300" or 30 Cal. however the groove to groove dia. between the US and Russian rifle can vary greatly. My particular Mosin measure .300" across the land and .312 across the grooves
Your US made rifles like the 30-06 and .308 use a bullet that measures .308 in dia. while your Russian 7.62 x 54 and 7.62 x 39 will be loaded with bullet that run between .310 and .310.5 it possible that you may find surplus ammo or new imported ammo that is loaded with slightly larger or small bullets at times.
The 7.62 x 39 falls closer to the 30-30 ballistics and is more of a short to med range cartridge while the 54r is similar to the 308 and is better suited for med. to long range.
This site has pretty much about anything you want to know about the Mosin family of rifles. http://7.62x54r.net/