To make it brief, the Fireside Poets were very traditional in terms of content and style, while Walt Whitman was far more revolutionary and experimental in how he composed his works.
The Fireside Poets achieved great popularity during their lifetimes, becoming the first American poets to do so within both the United States and Europe. However, their popularity declined with the coming of the twentieth century and the arrival of modernism. By then, their poetry seemed old-fashioned and trite.
Walt Whitman's works are more radical, both in their content (such as an honest sense of sexuality in Leaves of Grass) and style (his use of free verse). He wrote about subjects the Fireside Poets never bothered with, such as the common American person and the democratic spirit. Whitman's poems caused some bafflement upon initial publication, but the modernists of the twentieth century adored him, revering him as a uniquely American poet.
The Fireside Poets' poems usually had morals to convey or told stories of prominent figures, such as Paul Revere in Henry Wadsworth Longfellow's "Paul Revere's Ride." That poem has a traditional rhyming scheme and structure and details a great event in American history. It is catchy, traditional, and easy to digest for most readers.
"Song of Myself" from Whitman's Leaves of Grass is another matter. Whitman composes it in his signature free verse style and uses simpler language, a move that is radical but also appealing to a wide audience, challenging them without alienating them. Like the Fireside Poets, Whitman does indulge in some romanticism, glorifying the natural world, but instead of revering some great man or woman, Whitman is focusing on ordinary life, even at its most unpleasant, in his work (see the way he describes a biracial slave being sold and a drunk passing out nearby).
Whitman's narrator is also interesting in how he defies easy categorization. In Longfellow's "Paul Revere's Ride," the speaker is clearly a sort of authority figure, beginning the poem with the declaration, "Listen my children, and you shall hear / Of the midnight ride of Paul Revere," evoking the image of a parent or grandparent speaking to children. Whitman uses "I" to refer to the narrator, but many have argued the narrator is not simply Whitman, but a more transcendent figure:
I am not an earth nor an adjunct of an earth,
I am the mate and companion of people, all just as immortal and
fathomless as myself,
(They do not know how immortal, but I know.)
Every kind for itself and its own, for me mine male and female,
For me those that have been boys and that love women,
For me the man that is proud and feels how it stings to be
For me the sweet-heart and the old maid, for me mothers and the
mothers of mothers,
For me lips that have smiled, eyes that have shed tears,
For me children and the begetters of children.
Though the Fireside Poets' work is more audience-friendly in that it is easier to immediately digest, Whitman is going for mass audience appeal in his own way, trying to speak to and of American society as a whole in his own unique way. And this is why his work is still discussed today, while the Fireside Poets have become more of a historical footnote.