Jane Austen wrote this work around 1800, on the cusp of two cultural movements : Classicalism and Romanticism.
Elinor represents charecteristics associated with eighteenth-century neo-classicalism such as reason.
She demonstrates propriety, economic practicality and perspective.
In Ch. 1 we are told Elinor possessed a strength of understanding, which qualified her though only nineteen, 'to be her mother's coucillor.'
Elinor is probably a tribute by Jane Austen to C18th Classicalism.

In contrast, Marieanne represents qualities associated with the emerging 'Cult of sensibility' of romance, eager in her sorrows and joys, excess with no moderation with an admiration for the beauty of nature.

Marieanne and her mother nurture strong feelings...or emotion.
Their grief at Henry Dashwod's death is 'voluntarily renewed', as they 'gave themselves up wholly to their sorrow, seeking increase of wretchedness in every reflection'. (ch.1)

'Sensibility also involves the display of emotion to the exclusion of social propriety and feelings of others'.

-R.Gill and S.Gregory. Mastering the Novels of Jane Austen. (Palgrave.2003) p.73

Marieanne reminds us Jane Austen was a contemporary of late C18th Romantic poets such as Blake, Byron and Coleridge.
Yet S&S is not a straightforward study of contrasts.
Elinor, though representing sense does not lack passion 'She had an excellent heart, her disposition was affectionate, her feelings were strong but she knew how to govern them'
While 'Marieanne's abilities were, in many respects, quite equal to Elinor's. She was sensible and clever.' (ch.1).

Marieanne is probably her most Cowperesque when she wanders alone at Norland sighing in melancholic sensibility over it's trees and leaves.
It has been suggested Jane Austen was mocking excesses of the Cult of sensibility here and Marieanne shows self-indugence, yet she realizes nature is not responsive to her emotions, 'No leaf will decay because we are removed'.
I think Austen also deliberately gave Marieanne interesting qualities, such as a musical talent and literary awareness.

This is my first post for my focus of The Romantic Movement so I hope I've posted in right place. :)

Below, for interest is a link from the BBC about the Romantic poets of JA's time where you can read some of their works.