No there is no hidden advantage.
Looking at the weapons in question, the flail is 5gp cheaper than the warhammer, the morningstar and war pick are the only d8 one handed piercing damage type weapons so that might be something, but between them always go with the war pick since it is only 5gp vs 15 gp.
But the difference of a few gold is nothing in the grande scheme of things.
Piercing damage doesn't do anything extra I think to any known monster, so yeah nothing there either.
The weapon list and armor list is probably one of the worst things about 5th edition.