What's the difference between vacation and personal time?

Nothing (except in how they accrue, but that's getting very specific). Personal time originally was offered in place of sick days and other floating religious holidays. But Seattle now requires paid sick days, so you now get personal days + sick days.

You can't be sick for more than 6 days? What happens if you break a leg?

I wouldn't expect a broken leg to keep a programmer from working, but if it did then you'd have disability leave, probably paid for by your work via disability insurance.