Why does it matter in practice? Well, there's more than one reason, but consider that not every user agent is a browser with a person sitting in front of it. Your website also should be interpretable by content indexers like search engines, accessibility devices like screen readers, and so on.
Some services don't fit this model and really are better off being designed like desktop applications written in HTML and JS. But in my experience, most services can be modelled more like websites without making the design any more difficult to reason about, and almost all users' experiences are bettered by it.