To me, handwriting is an outward expression of inward feelings. My handwriting is usually very rushed and cluttered. (Though hopefully still legible!) It becomes very sharp and succinct whenever I'm angry. If it's the same for everyone, you could probably tell a lot about someone from the way they write.
Beyond that, handwriting is important for communication. Sure, verbal communication is largely electronic and oral nowadays, but if you need to convey a handwritten message and it's too messy for someone else to read, you're screwed!
You'll need to know how to sign your name for legal documents, too, so.
Eh. It's not -the- most important thing in the world, but it's something that requires at least basic skill.