It's not about brand. If it's a web application, call it a web application. If it's a website, don't call it by a wrong name just to attract customers by a fancy terminology. If you're selling Cessna, don't tell it's a space shuttle. Also, "application" is not a wider term. Applications and websites are mutually exclusive.
It's more subtle than the difference between static and dynamic content. A website which just displays some content from a database and lets the users to submit a contact form is still a website, not a web application. If I can just submit a feedback, well, I wouldn't call it an application. It's rather a website with a feedback page. If the website contains news section which is generated dynamically, it's still a website, despite the dynamic property of its content.
A web application is much more complicated, and mixes input, dynamic content and is conceptually perceived by people as an application.
Compare a web application to a desktop application:
- GMail for example is a perfect example of a web app: you can easily imagine a desktop version of it.
- An e-commerce website is still comparable to an application: even if you get all the data from the web, you can still imagine using it as a piece of desktop software product.
- A website which just says a few words about how a company is great, and let you contact this company through a contact form is not suited for a desktop version: you'll just not be able to use it as an ordinary application.
Is there more discussion about apps than about websites? Well, yes. In 1995, it was difficult to imagine a complicated web application. AJAX was not very known. Connection was generally terribly slow, and the browsers - terribly creepy. The model was to sell software on material support: floppy drives, CDs.
Today, we mostly have a decent, and sometimes quick and unlimited bandwidth, high quality and hugely fast browsers with hardware acceleration, etc., which makes web applications most popular. The new model is to sell software over internet, sometimes as a service, i.e. as a web app.
The evolution of the internet and the technologies and the birth of the most successful web applications (GMail among others) made it more and more interesting and easy to develop them. At the same time, websites can be done with WYSIWYG editors by people with no technical background: extremely badly, as usual, but done.
That's why companies prefer talking about web apps, not websites: they show that they are up-to-date, that they are working on hard stuff, and all this marketing bla-bla. Now, if they are talking about a web application when actually they are requested to make a visual design and paste some static HTML and CSS on it, they are using, intentionally or by ignorance, the wrong terms.