Because the people you want most aren't going to waste a ton of time jumping through hoops? The nice thing about puzzles is that they're self-contained and can be understood and solved in a few hours. When I was applying for a Facebook internship, I wound up spending ~6 hours over a couple of days finishing two of the puzzles. This already seemed like a significant time investment. A problem that involved "a real world use case, with a code base to work off of" would necessarily take a lot longer to understand and implement. I was pretty busy that month, and if the puzzle questions had been more time-consuming I might have skipped Facebook and applied to other companies instead. (As it happened I wound up at Google)

Puzzle-based interview questions are meant to be a rough cut to weed out the people who are clearly unqualified for the position. They're not meant as a substitute for more in-depth discussion of skills, work experience, etc. If the filter is too time-consuming, it's going to weed out some of the best people who are confident they'll get offers from other companies. And a large company like Facebook doesn't necessarily want to test specific skills because they may not have yet decided which position you'd be hired for, and in any event you're likely to work on multiple projects during your time with the company. For entry-level positions (which is presumably what these questions are designed for), it's better to hire the smartest guy you can find and train him on specific technologies than to hire a guy who already has a specific skillset and then discover he's a one-trick pony.