In general, Longfellow and Whitman have little in common as poets. Longfellow was a traditionalist, using standard verse forms, regular meter, and rhyme. Whitman was an iconoclast, in most of his important verse rejecting both traditional formal elements of poetry and also expressing unorthodox ideas. That said, "O Captain, my Captain" is, from a technical standpoint, one of Whitman's most conventional poems. Instead of being written in the free verse he is best known for, this poem has the standard elements of meter and rhyme.

Though "Paul Revere's Ride" and "O Captain, my Captain" both are patriotic poems extolling American heroes (Paul Revere and Abraham Lincoln), and though both are written using the traditional verse elements of meter and rhyme, the two poems have little else in common. The first is about an episode in American history in which a man successfully warned the populace outside Boston that an attack on them (or, more specifically, an expedition that had the aim of seizing their stockpiles of weapons and ammunition) was imminent. The second is an elegy, a lament over the assassination of the president. The first is about the start of a war; the second relates an event that occurred (approximately in time) at the end of a war. The first describes events literally; the second, metaphorically, in likening Lincoln to a ship's captain, lying dead as the ship pulls into port. The first relates what occurred 85 years earlier:

Hardly a man is still alive
Who remembers the day and year.

The second tells of a tragedy that has just happened. Both poems, however, have entered and have maintained their place in the American consciousness. They are both a kind of folk literature and are recognized even by people who don't necessarily read poetry or like books. Both have the ability to evoke a gut-level feeling of patriotism. And it is on this level that both Longfellow and Whitman are typically most successful in their work.

I would say that one specific point of reference for both is that they have become staples of American History.  Both are used as points of nationalistic pride in terms of telling the history of the United States.  Longfellow's work depicts Paul Revere as an individual leader of the movement that took grave risks in alerting the Colonists that "the British are coming."  He is shown as a stoic figure who was able to risk his own personal safety of being in trying to reach a higher and collective goal.  In Whitman's work, Lincoln is depicted as the fallen leader who was able to guide the nation through its darkest hour.  He is shown as a force of moral and ethical integrity, one that embodies what it means to be a "leader."  I think that both contribute to the mythology of the United States History as a hopeful narrative of good triumphing over evil.  Some significant points of distance between both is that Whitman's voice is much more present in his poem than Longfellow's is in his.  Longfellow subjugates his own voice in deference to the subject matter, while Whitman makes no bones about the fact that he is in mourning and his honoring of Lincoln is through his poem.  In this light, there is much more emotion in terms of language and tone in Whitman's work than in Longfellow's.