Are weapons that are "effective against birds/beasts/slimes/lizards/undead" really any better?
From experience of many playthroughs, yes, it makes a difference. Such weapons do noticeably more damage to the relevant type of creature. You can see this by simply using it: since you fight so many things with each iteration of a weapon, you get to know the typical damage numbers that appear, and the damage done to appropriate creatures with an "effective against" weapon is noticeably higher than expected.
The exact math is not available anywhere, as far as I know.