This is a great question to think about, as both of these poems are famous for their Romantic poetry and their treatment of nature through their art. This of course most famously resulted in their joint book of poetry entitled Lyrical Ballads, which is still studied on Literature courses today. The key ways in which their views on nature differed are expressed in their respective poems in this volume. Coleridge sought to show the supernatural in nature as being something that was wholly natural, and Wordsworth sought to show the natural as something that was strangely supernatural. This is an important distinction to keep hold of.
If we look at two poems from these poets we can see how this operates. In "We Are Seven," by Wordsworth, the speaker accosts a young girl in the countryside who insists that even though all of her brothers and sisters have died, "we" are still seven in number. She does not see death as something that separates her from her dead siblings. Note the frustrated voice of the speaker:
"But they are dead; those two are dead!
Their spirits are in heaven!"
'Twas throwing words away; for still
The little Maid would have her will,
And said, "Nay, we are seven!"
There is something supernatural in what at first glance seems to be a very natural sight. In the same way, consider a poem such as "The Rime of the Ancient Mariner" and note how Coleridge explores nature in a distinctly supernatural way through references to the Polar spirit and the fate of the vessel. This makes us view nature differently and to see it as something that is at its core supernatural. The two poets therefore differ distinctly in how they try to present nature in their poetry.