The answer: A lot.
Your hair frames your face, it can put emphasis on certain features or balance out your face. The best way to explain the huge affect hairstyles have on your appearance is with some before and after pictures.
In the example above we can see Ryan Gosling's oblong face shape does not suit shorter hair at all. His features such as his ears become too emphasized and give an imbalanced and overall unaesthetic look. He looks a lot better with a slick, longer hairstyle.
In this example Johnny Depp's hairline does not suit a buzzed haircut at all, it makes him look much older and he loses the unique look he's known for. Could you even spot him in a crowd of people with the buzz cut?
Chris Evan's improved his appearance ten fold with a nice beard and a slicked side part hairstyle. In the before picture he looks like the average guy you would walk by and not notice at the office.
Robert Pattison makes the transformation from average British lad to heart throb all within a few months of hair growth.
Henry Caville's big jaw looks imbalanced with a short buzzcut, but with a longer look his face looks much more balanced and movie star handsome
. 
Christian Bale's Oval shaped face much better suits more hair length as oppose to shorter hair.
David Beckham can pull off pretty much any hairstyle but you got to admit, certain hairstyles look much better on him.
Matt Bomer's hairstyle change made him go from handsome movie star to middle aged dad.
Channing Tatum's new hairstyle adds much more personality to his look and makes him look way better.
Hope this helped you guys see that a hairstyle can definitely have a drastic affect on your overall appearance!