In addition to the things already mentioned here: Hosting companies have economy of scale. Because they are running hundreds or thousands of machines:
- They can afford to pay people to stand by on call 24/7. Do you want to get paged in the middle of the night? Incidentally, did you set up Nagios and something like Pingdom and a pager service on this box in your closet, or are you just happy to go down for hours at a time?
- They see all the common problems all the time. When you experience common problems -- e.g. database crashes and needs to be restored from backups, bad PHP configuration is eating too much memory on the box, DDOS attack needs to be parried with iptables -- you are going to spend hours reading instructions and applying them. Going to all that effort for the sake of one small box is kind of a waste. What you want is to amortize your skills over many, many boxes. That's what professional sysadmins do.
The first time you have to spend a couple of hours reading some FAQs in order to keep your box running you will burn up 100% of the money you saved by not paying for a hosting service.
Of course, if you're trying to learn what hosting is about trying to run your own box is educational. Though I'd still do that learning on a VPS or EC2 rather than struggle with a home network and a random home box.