Definately a shocked Secret Service would lock down tighter on a presidential candidate and president. I don't think Lyndon Johnson had the willingness to compromise that was needed for a peaceful solution of the Cuban Missile Crisis, and probably no Test-Ban Treaty as a result of that. It's also possible that he would have been wiling to secretly have Americans helping Cuban revolutionaries at the Bay of Pigs.
What would really change is the American culture of the 1960s. There would be a generation of babby boomers who are not drawn to public service by Kennedy. I don't know if Johnson was enough of a visionary to call for a NASA missions to the moon. Without Kennedy giving blacks, women, hispanics, gays the inspirition to fight for what they believed in (and a Southerner in the White House), there may have been a Civil Rights movement later in the 60s or maybe even the 70s.