What is the difference between an iphone 4s and an iphone 5s?

My iphone 4s just broke and my parents want to just replace my iphone 4s with another iphone 4s, but I'm trying to convince them to upgrade me completely by getting an iphone 5s. My mom asked me why an iphone 5s was so different from the 4s, but I couldn't really come up with any reasons besides the obvious reasons like the 5s has a better camera and is just cooler. If you could help me by telling what the more important changes are or just in general why I should choose the 5s over the 4s that would be great.

Added (1). If you could also avoid info about the appearance or showy things about it that would be good too because even though those things are cool, I don't think that type of stuff will be enough for my parents to let me get the 5s.

There's a finger print scanner that will unlock your phone and type in your password for you just by placing your thumb or finger on the home button

You get to pay $500 for another 2 year obligation with the iphone 5. You can't do that with the iphone 4.