Broadly speaking, Wordsworth focused more on the natural; Coleridge on the supernatural.
The two collaborated on several poems and anthologies, the best of which is The Lyrical Ballads. So says the Enotes critic:
The interplay between natural and supernatural imagery in the individual poems has recently been studied by Roger N. Murray and Susan Eilenberg.
Both were Romantic poets who focused on nature for inspiration and enlightenment. Both sought to write poetry that captured the natural rhythms of language, of the common man.
Specifically, Wordsworth sought man to be in a “state of nature” free from the bonds of class, government oppression, and religious dogma. The central theme of The Prelude the love of nature leading to love of man. So says an Enotes editor:
He sees a correspondence between the unspoiled nature of man and the naturalness of his environment. Romantic ideology of this sort underlies much of the contemporary environmentalist movement: the feeling that man ought to be in harmony with his environment, that nature is beneficent, that man ought to live simply so that the essential part of his human nature may conform to the grand pattern of nature balanced in the whole universe.
Wordsworth's concept of nature was idealistic when he was younger and grew more hostile and cynical as became older.
Coleridge said, "My endeavors should be directed to persons and characters spiritual and supernatural, or at least romantic."
In Lyrical Ballads, Coleridge says:
Poet who hath been building up the rhyme / When he had better far have stretched his limbs / Beside a brook in mossy forest-dell, / By Sun or Moonlight, to the influxes / Of shapes and sounds and shifting elements / Surrendering his whole spirit, of his song / And of his name forgetful! so his fame / Should share in Nature’s immortality, / A venerable thing!
Coleridge expresses his conception of poetry in terms of a synthesis of imaginative vision and of actual perception, of “outer” and “inner,” in other words, or of “object” and “subject.” His earlier verse usually had nature as the “object,” but later poems approach treatment of the symbolic, mythic, and general consciousness of the inner man. Myths of death and symbolic rebirth predominate. Historically, this conception of poetry, a synthesis of the individual and subjective with the concrete and objective, moves sharply away from neoclassic doctrines of a poetry as imitation governed by rules. To Coleridge, art is less imitative than “organic”; a poem is a growing unity, the parts related to one another and all comprising a whole. The “end” of poetry is pleasure, not instruction, and yet a poem tells a higher truth. A poem expresses “a more than usual state of emotion with more than usual order.”