The main difference between the brewer's yeast and nutritional yeast is that
the former still has yeast in it. Sounds like a paradox, but nutr. yeast is
actually yeast-free; only the non-active ingredients (vitamins, minerals,
protein, etc.) remain. Both are good for hair, nails and skin.
Both naturally contain B complex vitamins *except* B12; it is added to some
nutritional yeasts (the Red Star brand is one; check the label!). I've never
heard of zinc in yeast; again, some nutr. yeasts may be fortified with it.
Brewer's yeast (I'm not sure if you're talking about granules or cake; I've
only used cake) has a very strong flavour that some people love and others
can't stand. Also, if you're prone to yeast infections (sorry, guys, tune
out for a minute), avoid brewer's yeast.
So if you're thinking of using either as a dietary supplement, rather than
to brew something, stick with nutritional yeast. If you mix brewer's yeast
with some warm water and sugar, you'll get some surprising results,
especially if the mixing occurs in your stomach!