Honestly, there are differences. Do they matter? IMHO, no.

Most "stopping power" debates are talking about rifles that are vastly capable with or without lead bullets, and with the bans that have happened, the tech has moved forward. e.g. Barnes Bullets (which are copper) are considered a premium round, not just a premium non-lead round. And they aren't really overpriced.

WRT shotguns, I don't know this end as well, but the ammo market could use some forced innovation (the firearms market in general lacks innovation).