MSDE is a cut down version of SQL Server without any front end tools. Maximum database size is 2GB and max ram is 2GB.
If you are talking about small, low traffic web sites then you may get away with this.
SQL Server Standard edition gives you the front end tools and the database size isn't limited. Max RAM is 2GB.
Enterprise edition is way over the top for one box web sites. RAM is dependent on the version of Windows.
I don't like running SQL on a web server, personally I like a sacrosanct database server behind a firewall and front end web server(s).
Licensing for web apps is per processor.
I would uninstall your demo version and do a clean install of SQL server to avoid any possible registry problems.