Let's say Martin Van Buren doesn't run or isn't nominated as the Free Soil Party's candidate in 1848. Without him, New York would most likely have gone to Democratic nominee Lewis Cass, and with it, the election.
So instead of Zachary Taylor, we have a president in the White House who is militaristic, supports the filibusterers in Central and South America, favors popular sovereignty on the slavery question, took a hard line with regards to secession, and is for the annexation of Texas...basically Stephen Douglas beta.
Butterflies anyone?