I don't think Truman would have conducted the war much differently than did Alben Barkley: the two of them were largely on the same page anyhow. Perhaps Truman might have reined in MacArthur before he got to the suburbs of Pyongyang. Barkley's orders to stop to allow the North Koreans time to declare Pyongyang an open city, rather than losing it in battle, and thereby saving face to some extent, wasn't perceived as a sound move at the time but it proved advantageous at the subsequent peace talks; it also gave the opposition something of a reality check without embarrassing either North Korea or China. The negotiated treaty under Truman might have been different had MacArthur not been reined in, or had been reined in sooner.
The 1952 presidential election would surely have been different. The ticket of Barkley and Joseph Kennedy was perhaps one of the less well-advised in terms of choosing a running mate: Kennedy had too much baggage from the late '30s and early '40s to offset the balance he brought to the ticket. Many pundits blame that baggage, along with the rumored marital infidelities (Kennedy was linked romantically with Gloria Swanson, among others), as the causes of the Democrats' loss to Dwight Eisenhower and Harold Stassen: the latter ticket was perceived as a clean change in administration, never mind the quasi-success of Korea.
The prime fallout of the Truman assassination was a prevailing feeling that the US should cut Puerto Rico loose, and be damned to them: not one cent for foreign aid; strip Puerto Ricans of all citizenship rights and deport them immediately. That policy, carried through Congress on a wave of anti-PR sentiment, was implemented in late 1951. The corollary-increasingly strong ties with Cuba, including aid to the reform regime of Raul Castro, brought Cuba into the US orbit once and for all: to this day, Cuba is not quite the 51st state but might as well be for all intents and purposes. Meanwhile, the Republic of Puerto Rico has suffered over the years from multiple revolutions, and perhaps the first amphibious war between smaller nations: Trujillo of the Dominican Republic saw an opportunity in 1955 to extend his reach in the Caribbean and declared war over a minor diplomatic pretext. The attempt at conquest was unsuccessful, but it did more harm to Puerto Rico than to the Dominican Republic. As of this writing, Puerto Rico is the poorest of the Hispanic Caribbean nations, and is likely to remain so for some time to come. American diplomats consider San Juan to be the equivalent of the US Marine Corps' "motivational platoon"; i.e., duty in San Juan is viewed as no better and quite possibly worse than that in any number of sub-Saharan nations.