Laptop as a web server?

Remember to compare apples to apples.
The laptop would be a dedicated server but I suspect you're comparing it to shared hosting plans.
The downside of the self host option would be your internet connection upload speed.

If you go the laptop route I personally would opt for formatting it and putting CentOS or Debian on it then Nginx instead of Apache, then MySQL and PHP, perhaps Varnish too. Nginx and Varnish should help a bit with performance to help counter a slower uplink speed versus a real host. You can put a free cPanel alternative on, good article on lordmatt.co.uk which lists plenty. You can use the dns management on the server once set up to create mx records to point to google for your email if gmail accounts allow you to use your own domain.
 
The price of hosting these days isn't worth setting up your own. Fair enough if you were learning play about with a vm or something like that but hosting on it's own is that cheap there is no other answer.... purchase some hosting space and domain name and host it somewhere else.
 
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