For some genres, procedurally-defined levels are fine, but when handcrafted levels are done properly there's a pretty substantial difference. I'm reminded of an interview[1] with Shigeru Miyamoto and Satoru Iwata discussing the first level of Super Mario Brothers:

  "But if you avoid the first Goomba and then jump and hit a block above
  you, a mushroom will spring out and you'll get a shock. But then you'll
  see that it's going to the right so you'll think: 'I'm safe! Something
  strange appeared but I'm okay!' But of course when it goes against a pipe
  up ahead, the mushroom will come back.

  At that point, even if you panic and try to jump out of the way, you'll
  hit the block above you. Then just at the instant where you accept that
  you're done for, Mario will suddenly shake and grow bigger! You might not
  really know what's just happened, but at the very least, you'll realize
  that you haven't lost the turn."

[1] http://us.wii.com/iwata_asks/nsmb/vol1_page4.jsp