What are the basic differences between Wordsworth's and Coleridge's philosophies in their respective works?

The question of where Williams Wordsworth and Samuel Taylor Coleridge differ in opinion on the correct understanding of poems and poetry as expressed in their respective writings Preface to Lyrical Ballads (1800) and Biographia Literaria (1817) is a complex one requiring a detailed answer.

In this forum, an overview is the only kind of answer possible, but it will lead you toward further research. Most of Coleridge's rebuttal to Wordsworth's Preface is in Chapters XVII and XVIII of Biographia, with definitions of poem and poetry in Chapter XIV.

Colridge's disagreements with Wordsworth lie in the following points.
(1) What constitutes the language of "real life."
(2) Whether the feelings of lesser educated farmers (of the 1700s) are more genuine than the feelings of people in the higher classes.
(3) Whether Aristotle's principles of poetry do or do not govern poems and poetry.
(4) What is to be done about language that is dull and garrulous in everyday conversation, which is still dull and garrulous in poems. Wordsworth cleans it up, so to speak; Coleridge says don't use it because it has to be cleaned up, which renders it the same as the language of higher classes.
(5) That Wordsworth's poetry about "low rustic" people, feelings, ideas disproves his own philosophic assertions because of (4).
(6) That rustics focus on facts while individuals at higher levels of education, work and experience focus on the connections between facts and the connection of facts to governing laws, hence focusing on a higher order of thought, ideas, language, conversation.
(7) These support Coleridge's chief point that there is a difference between the language of prose and poetry, while Wordsworth asserts that there is no difference.

Breaking with the earlier eighteenth century, which maintained that poetry should be rational and objective, Wordsworth's "Preface to the Lyrical Ballads" focused on the subjectivity of the individual. He writes, "Humble and rustic life was generally chosen, because, in that condition, the essential passions of the heart...speak a plainer and more emphatic language."  He believed that feelings "coexist in a state of greater simplicity" than "rational thought. Thus, to Wordsworth it is a poet's duty to capture and express experience with authentic and internal force, "a spontaneous overflow of powerful feelings with emotions recollected in tranquility."The poet should descend from his or her "supposed height" and "express himself as other men express themselves". This statement lies at the very essence of Wordsworth's theory of poetry.Notice, that in the preface to "Lyrical Ballads," there is not much about imagination.

Samuel Taylor Coleridge brought back the issue of imagination and fancy in his discussion of poetry in "Biographia Literaria." While to Wordsworth, imagination was something taken for granted, especially in poets, Coleridge, while agreeing with Wordswoth's premise, nevertheless, clarifies for himself what imagination was; what was the difference between imagination and fancy that the eighteenth century critics tended to merge together?

The Biographia Literaria was one of Coleridge's main critical studies. In this work, he discussed the elements of writing and what writing should be to be considered genius. The "Biographia" blends criticism of poetry with literary theory, philosophy of Immanuel Kant, and is full of discussions on politics, religion, social values and human identity.

Central to the Biographia Literaria," is his discussion of imagination and fancy.

Like Wordsworth, Coleridge, too, rejected that the mind was a tabula rasa on which external experiences and sense impressions were imprinted, stored, and recalled. Rather, he believed that imagination is innate. Coleridge divides the mind into imagination and fancy.

To Coleridge imagination was primary and secondary. Primary imagination, was the creative force behind perception itself; meaning, that there is nothing called objective perception. Perception, according to Coleridge, is essentially subjective. While the human being is finite, Coleridge maintained that the poet's creation of "I AM" is his or her expression of the infinite. What this means is, according to Coleridge, if we could remind ourselves of the "I AM," we can, through our writings, move gradually from the finite to the infinite.

FANCY, on the contrary, is much more limited. It comes from memory, according to Coleridge. When we free our memory from being to bound up with time and space, we are in the realm of fancy. Its provenance is memory and its interaction is through the association of ideas. Whereas imagination is active and dynamic, fancy is "passive and mechanical." Imagination, on the other hand, is "vital" and transformative, "a repetition in the finite mind of the eternal act of creation."e and inventive genius."

Frederick Engell, one of the two original Marxist's critics, has observed that Coleridge's division of the imagination into the "primary" and "secondary" draws a distinction between creative acts that are unconscious and those that are intentional and deliberate. Imagination works at the unconscious level, while fancy is willful and deliberate.

What are the similarities and differences between Wordsworth and Coleridge as romantic poets?

I would say that the primary similarity is that both poets embody many tenets of the Romantic mode of composition.  They both valued the subjectively emotional and personal experience.  Both of them also felt that the construction of art suffused with an affect point of view is of vital importance for both the artist and the reader.  I think that a significant divergence is that Wordsworth was more concerned with what seems to be a philosophical expression of what Romanticism was through literature.  Coleridge certainly feels this is important, as well, but I feel that Wordsworth devoted more works that help to better understand this particular philosophical expression of consciousness.  I would also suggest that the role of nature is one that might be different between them.  For Wordsworth, the role of nature is one that is more akin to being able to express aspects of one's reality that can be seen in the most simple of things.  A field of daffodils or the singing of a maiden's song in a pasture of grain contained aspects that one could use to explore one's own sense of self.  Colerdige depicted nature in a more supernatural or "other worldly" manner.