Web site, think blog, information landing page, or even web store product page(s).
Web app is highly interactive (may or may not be an SPA single page app), relying heavily on javascript for UI events, saving/fetching application state, remote datasets, and sometimes making use of modern HTML5 browser features.
Web site is displaying information and is just augmented with some javascript mostly for improving the UX or attaching analytics/social plugins etc.
Thus I am a complex website engineer. It sounds ridiculous. The webapp developer is domain knowledge wise closer to traditional "application development" than to traditional "website development".
In other words as someone else noted, it is a sandboxed application with
sandbox execution environment == browser IO == REST UI = HTML5 DOM manipulation of the same page
A website (or server app) is a series of HTML5 web apps or HTML5 static pages (JS-less web apps) with navigation logic and inter-app state living in a server app.
web sites are content centric and made for consuming it.
web apps are interaction centric and made to "do" something.
Note that this definition means that certain pieces of software could be considered either "sites" or "apps", depending on how they're being used.
For instance, blog software like Wordpress would be a "site" for the readers, but an "app" for the person who writes the posts.
One part is an editing software/gui and one part is a reading software/gui. Both working on the same data.
Since both are used by different kind of users, I would design them as different programs (one a site and one an app).
Web App are application like website.
So instead of traditional application/program such as Quicken we got Mint.com now. It makes application OS agnostic now too.
Web sites are bunch of pages that are hot link or connected via URL. Think of the old days of geocities, xoom, tripod, etc..