What are some pitfalls of hosting a website from home?

• Electricity cost vs. Hosting cost
• You usually don't have much redundancy at home (i.e. only one server with no automatic failover)
• Cost of making the setup redundant vs. Hosting Cost
• You need to be careful to separate your Server from your home network, otherwise security issues in your server can lead to a breach of your private network

First and last are the big ones. Most ISPs disallow it per their terms of service, and some may even block inbound port 80/443 traffic. Remember too, many broadband connections are setup to allow much greater downstream traffic than upstream traffic (what was ADSL in the DSL days). So you may get 15MBit down, but only 4MBit up. If you're serving that means you've got a 4MBit choke point.

• Environment issues. (Dust, vibration, storage space, do you have a suitably out of the way place for your server to live?)
• Heating and Cooling issues. (Operating a dedicated server can add significant heat. Also, your server may overheat during a heat wave if you don't have adequate ventilation.)
• Security issues. (Your server could be a conduit allowing access to your private data and computers.)
• Network Demands. (You have to be more careful how you use your network. If you use too much of your connection (e.g. for downloading) you may inadvertently DoS your site. Also, you have to be more careful about planned downtime for your router or network, you can't just reflash dd-wrt on your router on a whim (for example), as this could bring down your site when someone is using it.

For purely personal, low-traffic sites, I really don't see it as an issue. In fact, I'd recommend it as a starting point for a purely hobby type site. If your site gets popular, then look at migrating to a hosted setup.

On the other hand, if its any sort of business site, run it on a hosted provider, presuming the business isn't in the business of hosting (or big enough to own its own data center).

Something no-one seems to have touched on yet: the psychological effect.

When I ran a server from home (which acted as my domain's mail server), I found I was constantly paranoid about it staying up. I was always pinging it to make sure that it wouldn't drop that potentially important e-mail. On the occasions that I couldn't access it, panic set in; has the house just been robbed? burnt down?

Of course, you may be far less prone to such thoughts than I ;)