I could easily change things around a bit and ask if we should colonize Mt. Everest (the world's tallest mountain), the Mariana Trench (the world's deepest ocean) or Mt. Vesuvius (a volcano). All three would teach us a huge deal in many fields, but is, apart from that, about as pointless as colonizing the Moon.
The only reason why everyone are so crazy about colonizing some dead rock out in space as opposed to some dead trench or mountain or some active volcano is the "cool factor". It's not that we'll learn this or advance that way. It's that space travel fascinates. The abundance of sci-fi movies, games, and television series proves as much. It's not about learning, advancing technology, or "finding out how we came to be here". It's about having fun.
Look, we'll get man to Mars in time. We'll estabilish our colonies in the Andromeda Galaxy. We may even one time in the far future have grown to tens of thousands of billions of people and spread our colonies, explorers and probes to all over the galaxy. What's the bloody hurry?
Again, if you've got a moderately serious illness, what do you do? Do you spend your money on that cool, awesome PlayStation 3 you want so much - or do you buy antibiotics and save the console for the next time you've got the money?