Metro Style App was a working name before Microsoft settled on Windows Store Apps.
Windows Store Apps just like desktop apps run on local machine, but are often architected in more modern ways where the app would often connect to web services or push notifications that run in a data center. The UI though usually runs on local machine even if it is written in html/js with winjs.
The biggest difference is distribution - web apps run on the server so you only need to navigate to them in your browser to run the latest version. Desktop apps need to be installed - either with xcopy-style install, Windows installer or perhaps the ClickOnce technology. Store apps are distributed through the store which makes them easier to install and update than desktop apps as well as monetize, but are sandboxed (have limited access to local machine resources) for security.