In my mind, the primary difference between both was the view of society as well as individual expression. Neoclassicist poets were more concerned with the notion of a collective entity or social view of the individual, whereas the Romantic were more driven by a sense of uniqueness to the individual and separation from the social order. Whereas the Romantics saw the Neoclassicists as striving for a sense social conformity, they viewed themselves as seeking to liberate the individual from these constraints. Additionally, there was a greater emphasis on technique and adhering to these poetic forms for the Neoclassicists, whereas the Romantics were focused on poetry and art being "a spontaneous overflow of emotion."
The neoclassical poets like Dryden and Pope primarily deal with the intellectually constructed satirical genre and approach poetry from a highly elitist point of view where rationality and decorum become more important than anything. They follow nature and compose a kind of allusive objective work of poetry, urban, cynical and nonchalant. Wit and a highly prescriptive sense of discipline govern their poems. The Romantics like Wordsworth and Coleridge, on the other hand de-elitize poetry, trying to restore to poetry the democratic spirit of day-to-day language, staging a return to nature, prioritizing emotion and imagination over reason and decorum and opting for the lyrical or subjective poetry of self-expression. They value the power of spontaneous poetical creation more than the prescriptions of a fixed literary genre.