Even though both authors are the founders of New American Poetry, a movement which pulled away from the Romantic authors' strict structure, Whitman and Dickinson write very different poetry. While Whitman often writes in free verse with no standard meter or rhyme scheme, Dickinson follows rhyme patterns and usually a very strict meter (almost all of her poems--because of their meter--can be sung to the tune of "Gilligan's Island").
Another difference is that Whitman stresses an undeniable optimism in mankind. He follows Emerson's view that man cAN be self-reliant and find answers within himself. His "I Hear America Singing" praises the common worker. In contrast, Dickson's poetry is often very personal and discusses inner struggles. Her "Heart, we will forget him" demonstrates this theme, and much of her poetry seems unconcerned with one's role in the community (most likely because of her reclusive nature).
A similarity between the two poets is that both veered away from the Romantic poets' traditional view of God or a Higher Being. Whitman stresses an Emersonian Oversoul or a "god within one's self," while Dickinson is more concerned about earthly emotions and relationships. Admittedly, both poets include aspects of spirituality in some of their poetry, but those spiritual elements vary drastically from the Romantic poets' themes.
The previous answer is quite funny. I think that there can be many points of convergence and divergence between both. The primary point of similarity in my mind between both thinkers is a praising of the subjective experience. This idea was of vital importance in the writing of Dickinson, who was driven to explore the different valences of the individual and an existing social order. At the same time, Whitman was part of the Transcendentalist Movement, which located the subjective experience as essential to understanding the nature of truth and the individual's quest for it. Whitman was firm in his belief that the American Historical experience and its political expression of democracy represents something worthy of praise. Dickinson was more personally driven, more subjective, than the expression of this political ideal.
Whitman knows how to use punctuation. Sorry, I couldn't resist.