Hefeweizen is a perfect example. The yeast produces particular esters/phenols reminiscent of banana and cloves. If you 'cheat' and use a neutral dry yeast, you're going to end up with a plain wheat beer that is NOT a hefeweizen.
Hefeweizen, Wit, and most belgian styles are probably the most prominent examples where appropriate yeast is crucial. There are specialty yeasts for english, american, german, etc beers but in many cases you can still get in the ballpark of the right style with a general neutral yeast.