Nixon campaign is a little smarter (no 50 state strategy), he wears makeup in the debate, and narrowly wins the debate and the election.
Bay of Pigs probably happens as it is approved under Ike, but it isn't as big of a disaster as it was under JFK, and Khrushchev respected Nixon and wouldn't see him as an wimpy novice like he did Kennedy, so there's no Cuban Missile Crisis.
I see Nixon responding to the mild recession JFK presided over OTL with a Tax cut just like JFK. Democrats probably make gains in '62 as that would be 10 years of Republicans in the White House.
The '64 election most likely goes to Nixon. He defeats LBJ or Symington or another Southern Democrat, as I can see the Dems nominating a Southerner to the ticket after northerner Kennedy lost them the 1960 election.
Civil Rights and Voting rights happen. In OTL, Nixon did not oppose those policies pre 1968, and even in 1968 he was just pandering. Whether it happens in 1964 depends. It could happen earlier depending on how bad Nixon wants it and how willing LBJ is to get the votes in the Senate. It could happen later than '64, but not later than '68 as Nixon given the demonstrations and Protests would have to address the issue and take action on it.
1968 goes to the Democrats, and if a southerner loses to Nixon in '64, Humphrey, Muskie, or if he gets into politics TTL, Bobby Kennedy ends the nominee in '68. I can't for the life of me see the GOP hold the White House for more than 4 consecutive terms (I find 3 to be a bit of a stretch) in the New Deal Era, which was 1933-1969 OTL (or '33-'81 depending on your interpretation).
The big question for me is Vietnam. Does the war happen if Nixon wins 1960? On the one hand Ike sent advisors there and Nixon did escalate it during his Presidency of OTL after campaigning in '68 to end it. On the other, Ike had no intention, as far as I know, of escalating the conflict and sending troops in, and it was Kennedy who sent in the first bunch of combat troops, and LBJ who had up to 500K troops there and made it the quagmire that it was. So maybe Nixon doesn't get involved. Maybe he doesn't. My guess is one of you have a better idea of whether he would or wouldn't have.
When all is said and done, Nixon is remembered, so long as there's no scandal, as a pretty solid President, if not, dare I say a great or near great one, or he's seen as the Bush 41 to Ike's Reagan, especially if he somehow does lose in '64. Worst Case, he's remembered as he is OTL.