Though many teachers, professors, and experts like to lump Whitman and Dickinson together into the category of Transcendentalists, I prefer to teach Whitman with Transcendentalists Emerson and Thoreau and to teach Dickinson as a poet who does not necessarily fit any particular literary period. While Whitman (a bit) more strictly writes his poetry as reflections of the Oversoul (God, nature, and humans are all connected), Dickinson values nature, but does not really include any other tenets of Transcendentalism in her writing.
Both Whitman and Dickinson were known for being social outcasts, in a way. When Walt Whitman first published his most famous collection of poems, Leaves of Grass, critics and common American audiences thought it "crude" and "vulgar". We surmise now that he was probably also bisexual, though there is no solid evidence of any same-sex relationship in his life.
Emily Dickinson, on the other hand, lived a very solitary life -- out of choice -- and some experts say she only left her house to go on trips six times during her life. Reclusive and misunderstood, she never married; after she died, her sister found hundreds of poems strewn about her room, even stuffed in desk drawers.
As far as their writing styles were concerned, they marched to the beats of their own drummers (a phrase adapted from a famous Thoreau quote). Whitman was one of the first American poets to use free verse, an unrhymed, unmetered type of poetry; and Dickinson was one of the first to use unconventional punctuation (such as dashes) and capitalization. Many critics hail both Whitman and Dickinson as the all-time "greats" of American poets.
A few of Whitman's and Dickinson's personality differences translated into differences in their poetry, however. Whereas Whitman was quite outgoing and even spent part of his life volunteering to help wounded Civil War soldiers, the extent of Dickinson's generosity involved her lowering mid-afternoon treats out of her bedroom window for the neighborhood children to enjoy. Whitman's experiences with the wounded soldiers led to some thought-provoking writing, starting with his journals and ending with some heart-wrenching poetry. Once his "Captain" of the Civil War, Abraham Lincoln, had died, Whitman wrote his famous tribute in the form of "O Captain, My Captain!" Dickinson's typical lack of human interaction may have contributed to the overall depressed tone of her poems, with the notable exception of "Hope is the Thing with Feathers". Also, Whitman's poems quite often proudly blazon his individuality ("I celebrate myself, and sing myself" -- from Song of Myself), while Dickinson's quite often make more blanket statements about humankind in general ("Demur -- you're straightway dangerous -- and handled with a Chain" -- fron "Much Madness is Divinest Sense").
I hope this answer helped you. If you need more help understanding Transcendentalism -- or any of the other American literary periods, for that matter -- just let me know!
The previous post was quite thorough. I would only like to add that one particular point of convergence in both is how the notion of American literary voice was rooted in self expression. Both thinkers held true to the idea that any notion of the universal comes from the subjective, and that from this only can truth be fully understood. However, within this form of expression might also be a point of divergence. Whitman is quite passionate about the democratic political form as being the best political structure to express this subjective experience. Whitman is able to assert complete confidence in how the heterogeneous composition of American Democracy helps to enhance individual voice. Dickinson is not so sold on the idea of political expressions of the good, in general. Her writing does not explicitly articulate how politics fits into the subjective expression of self. This difference in perception on the role and function of political orders might be one additional area of contrast between both thinkers.