I think of Austen as basically a "classical" thinker/spirit witnessing and responding to the rising tide of Romanticism, especially as it played out in the emotional and imaginative lives of young women...the avid young readers of Cowper and Scott, taking Romantic poetry as a filter through which to process their own experiences.

Austen creates "Romantic" characters like Marianne and her mother, but maintains a certain distance from their "Romantic" mindset. As a narrator, she describes (or refers to) their raptures and storms in a detached tone, and she often shows them to us through the point of view of more self-disciplined characters like Elinor, Col. Brandon, and Edward. (An earlier post made that great point about Austen's "free indirect" narration, shifting frequently from one p.o.v. to another within a scene).

It's not that Elinor et al. don't feel strongly, but that they exercise control over how they express and act on their feelings - more in keeping with a "Classical" style.