What is the difference between a Start-Up and a small company which is not a Start-Up? What exactly makes a small and youg company a "Start-Up", among other small and young companies which are not necessarly start-ups?
Probably, there's no clear-cut definition, however, there's an article of Paul Graham called Startup=Growth which represents the opinion many agree upon: startup is a company that plans to grow fast (... and scale). A quote:


  A startup is a company designed to grow fast. 
      Being newly founded does not in itself make a company a startup. Nor is it necessary for a startup to work on technology, or take
  venture funding, or have some sort of "exit." The only essential thing
  is growth. The only essential thing is growth. Everything else we associate with startups follows from growth.


That's why a new barbershop isn't a startup since


  A barbershop doesn't scale.


This definition probably still leaves some space for interpretation, but not that much.