Bamboo Steamer VS Normal Steamer
The only difference between a bamboo steamer and a metal/plastic steamer is that a bamboo steamer will absorb (some of the) moisture from the steam, rather than allowing it to recondense and drop into the food.  

It's possible that recondensed moisture could take a small amount of water-soluble nutrients with it, but between the limited ability of bamboo to absorb water and the limited amount of time that water would be in contact with the food, I don't imagine it adds up to much.   

What it may add up to is a slightly different cooking process in total.  A metal steamer gets a bit hotter than a bamboo one, meaning the bottom of the food might be a bit crisper.  That can be useful, or not - making dumplings or other bread products, lower heat means less sticking to the wrapper/steamer.  A bamboo steamer will absorb some of the flavor of what you're cooking (and keep it), meaning it can add some interesting flavors to other things you cook (which can be good or bad), although that's controversial and appears to be only a minimal effect.

I tend to stay away from plastic steamers, as even if they are BPA-free, who knows what the next BPA will be.