Faster A7 64-bit CPU: Apple says the iPhone 5S is twice as fast as the existing iPhone 5. Since the iPhone 5C runs on the same processor as the 5, it stands to reason that the 5S will be twice as fast as the 5C. That seems like a big difference. (Check out Stephen Shankland's article on Apple's A7 chip to get a better understanding of what the 64-bit architecture means).
Better camera: Apple is touting how good the new camera is in the 5S and the better flash performance (you can read all about the 5S' camera here). With the 5C you get the iPhone 5's camera. Also, because the 5S has the faster A7 processor, it enables such features as slow-motion video capture. That feature isn't available in the 5C.
New M7 chip: Inside the 5S is a new M7 chip that's designed to be paired with next-generation fitness-tracking apps. It appears your phone will also double as a Nike Fuel Band. Those things cost $150, but who's counting?
Touch ID fingerprint scanner: One of the 5S' key upgrades over the 5 is the new fingerprint security system in the home button. We can debate about how cool and useful that feature is, but it's not in the 5C.
A lighter phone: The iPhone 5S (and iPhone 5) weighs 3.95 ounces; the 5C weighs 4.65 ounces.
An extra year of future-proofing: After a few years, Apple typically stops supporting certain devices for its OS updates. A few years from now, when Apple's releasing iOS 10 or 11, you may not be able to update the iPhone 5 or 5C with it. But you'll be able to update the 5S.
Better resale value: With Apple's new trade-in program, in a couple of years, if you go to trade in your 5S (or sell it on eBay or Amazon), you're going to get more for it than your 5C. Maybe as much as $50 or $100 more.