It would differ so much from Nixon, I'm surprised by some of the answers in this thread saying otherwise. I'm going to try my hand at a mini-TL on a few major differences I could plausibly see occuring.
Robert Kennedy would push very hard for constant and continued funding of NASA, with huge potential impacts. The Moon Landing was basically the easiest political slam dunk in Presidential history. It will be even moreso here. In this TL the slam was thrown to Robert by his tragic, darling, forever young and handsome brother, the late President John F. Kennedy.
Robert Kennedy is a smart politician whether you agree with his positions or not. He is also incredibly media savvy. He will firmly plant a huge portion of his Presidency and legacy and most importantly the "Kennedy name" into the Space Age. NASA will be having innumerable OTL cancelled projects funded in it's future which could have huge knock on effects. Short-term RFK will press for higher budgets (and more importantly, more photo-ops, discoveries, and positive buzz) for NASA at every opportunity. I would expect NASAs budget to increase significantly every year of his Presidency because of this mutually beneficial relationship.
Vietnam will look very different. RFK will not escalate to Laos/Cambodia. Would he take a hard-line, commit more troops, bungle it just as bad? I'd doubt it. Nixon was independent and had his own strategy but he mostly followed the generals/Kissingers bad advice. Vietnam will likely have a peace conference sooner. RFK will want Vietnam to be associated with the Johnson Administration as much as possible. He will attempt take all the credit for the peace and none of the blame for the war. A sizable force of U.S. advisors and special forces stay on after the full withdrawal. A continued air campaign and financial/military aid for South Vietnam is also provided for. Depends on butterflies but full withdrawal likely happens in late '71 or early '72. Again, this is just good political optics. Getting most soldiers home before Christmas would be a huge boon to RFK and the Democrats in the upcoming election. Whether Vietnam fares better than OTL will be a question for after the election which is how President Kennedy would like it.
For now he can run on peace, Space, and his signature domestic accomplishment, single-payer health-care. The latter two are both personally sweet to the President considering how instrumental both his brothers, John and Ted, were in their respective creations.
So what if you are the voter here? What would you do in '72.? The Kennedy media machine is splashing the pictures of President Kennedy triumphantly signing single payer health-care law on the Resolute Desk. Ted is beaming, teeth as white as the Oval office, standing just off to the side. What a triumph for family and America! Something that you can easily sell to voters. All of this captured on the most modern cameras and broadcasted into the exploding medium of cable news and television. The still relevant newspapers are already buzzing about Ted's future. A third Kennedy in '76? Could the country survive such audacity? Was 3 Kennedy Presidents just the right amount of absurd? Articles like that are front page right next to ones waxing poetic on how Jack's spirit was assuredly at the signing as well. Or how it was felt during the Moon Landing. Or the official peace conference in Paris (Jackie reluctantly agreed to make an appearance even, if only to help distance her late husband's name from the war and onto the peace). Or whenever the voters heartstrings need an appropriate tug. Could this momentum turn the White House and Congress into a Kennedy family tree? Has it already?
I'm not sure how much this matters but Nixon was the last President to have an all-white/man cabinet. RFK likely has some integration on his cabinet, could maybe even be an important one if it polls well in the right places. Just another easy layup for President Kennedy and another history defining photograph he gets to be in.
Honestly, who is the poor Bob Dole who gets to go head to head with that in November? I'm not sure who it would be but I pity them. I also feel really bad for Nixon here, he probably can't even look at the letter K anymore. But he never has to be known as the only President to resign so maybe his ending is happier? LBJ gets the John Adams treatment. He probably goes out more or less as cranky as he did in our time.
Five SCOTUS seats came up in the 8 years Nixon was elected to. I won't get into the weeds here since this is already getting really long. But that will have ENORMOUS effects on the history of these altered United States.
Without Watergate and a different Commander-in-Chief will South Vietnam last until RFK is retired? What role would China play in this world? The Soviets? No drug war? How does the GOP come back from this? No Chilean coup-Pinochet never comes to power? Will K-Mart beat out Walmart due to K positive butterflies? So many butterflies of different events and their respective changes to the timelines! You can almost smell the copper if you think about it. You can disagree with everything else I've said, but c'mon it would be completely different.
TL;DR: Alot Much.