What are the drawback of not using "brewer's yeast" when brewing beer? When walking through a brewery supply store, you will see a variety of yeasts. 

Likewise, you can find a variety of yeasts when walking through a grocery store.

What "happens" if you use the 'wrong kind' of yeast when brewing? How much of a beer's flavor / texture / body / etc is derived from the yeast vs all other ingredients?
Short answer; you get a lot of the flavour from the yeast, you can get a fairly big difference by varying the yeast in your recipe.

For example, a Saison and a Pale ale have similar ingredients, but taste quite different, a Saison yeast imparts a considerable amount of flavour... likewise if you use a lager yeast, and properly lager your beer, you can get a different flavour again, without modifying any other part of your recipe.

Lots of discussion on HomeBrewTalk forums, and other places. For a fun experiment, brew split batches and try two different yeasts to compare!