No. They are the same exact primer except for the brass anvil (the little tripod you see upside down in the primer). As long as you have adequate firing pin energy you should be good to go.
TulAmmo has both the KVB-556M and the KVB-223M available as magnum rifle primers. The former are the NATO spec primers. I bought 5,000 of them and 5,000 KVB-762 primers last year when Graf & Sons had them on sale for $16/1000 in that quantity. What I've noticed is the labeling of the cartons of 1000 and the individual 100 packs don't have complete information on them. Only the outside carton of the 5000 gave the actual Tula Russia plant's labeling (that's where I picked up the hyphen after KVB) and showed what the exact primer was inside. If you got such a carton, that number might be on it. If you bought fewer than that, the labeling may not be telling you what you want to know. You might be able to call Wolf and ask which of those two numbers are actually in the boxes.
As near as I can tell, the KVB-762 and KVB-556M both are beating all the domestic primers I have for muzzle velocity consistency with stick powders. I haven't done a run-through with spherical propellants yet. These primers are a little harder to seat than I'm used to, so I run even my non-crimped brass through either my Dillon Swager or apply a Wilson primer pocket reamer to them to get maximum room in the pockets before using them.
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