With the classic web-head look, I think Ditko introduced the concept of the completely masked Superhero (at least at Marvel, if not for the Silver Age in general) Is there another superhero whose faced is completely hidden in a Silver Age comic at the time?
The only one I could come up with was Iron Man, and Spider-man precedes him by almost a full year. It makes for a striking visual with the exaggerated eyes and lack of mouth to give you insight into the character's thoughts.
While it might be arguable how much this fully masked designed played into the popularity of the character, what is important is how much it adds into the transformation of Peter Parker into Spider-man. It completely removes any signs of the callow boy we are first introduced to at the beginning of the story.
The full face mask is also is significant because it allows Lee and Ditko to compartmentalize the two characters a bit. Peter remains a young boy at odds with his world while the Web Slinger is the MAN. As Ditko himself says:
I wasn't sure Stan would like the idea of covering the character's face but I did it because it hid an obviously boyish face. It would also add mystery to the character....
This idea of the faceless character would be one that Ditko would revisit a number in his career, most notably with The Question:
Later versions of Dormammu would add a mouth and more expressive eyes, which I think was a bit of a mistake because it removed some of the mystique and unapproachability* from the character.
As to the design of Peter Parker himself, I think need to look no further than this classic self-portrait of Steve Ditko:
Have a great day!
*Yeah, I'm not sure unapproachability is even a word, so feel free to suggest a better one.