One important thing about RFK's run for the Presidency is that even with all his charisma and inherent goodness, he wasn't afraid of playing dirty. He's almost commonly today accepted that the Kennedy campaign publicly leaked the information on the Peace Talk Scandal, that Nixon was trying to delay the peace talks which led to end of the Vietnam War for political gain, a leak which ultimately destroyed Nixon's campaign in 68' and even got him in trouble legally leading to the investigation that eventually led to his arrest in 1970. Humphrey on the other hand is a person who I'm not entirely certain would have been so willing to leak said info, even for political gain. So a Nixon victory in 1968 is actually very possible, if not I think likely.
That itself is going to have various effects on the ending of the Vietnam War. Rather than us starting to pull back in 69' like we did in our timeline, we might end up staying in Vietnam a lot longer, as there are a lot of sources which indicate that Nixon might have chosen to stay in Vietnam even if he had won.