Some aspects of the Great Society were very successful but on balance it was a major factor in prompting the late '70s tax revolt (if I recall correctly, prop 13 in California was the first big anti-tax law—it limited property tax very low, and is a key reason Californian pre-university public schools aren't that great). From a big picture standpoint this anti-tax and soon anti-government assault was not terribly beneficial. Mostly because it meant that the Republican Party got infested with supply-siders whose key argument—lower marginal rates always good=tax cuts all the time, especially for rich people—obscured the finer points of economics. So while some marginal rate cuts are good (90 to 50, for instance) later on it their obsession became quite silly (going from 28 to 31 will destroy the US economy!!!). At the same time this also marginalized a lot of the old conservative groups: the Dole and Ford Republicans who were solidly conservative but not irrationally so… they followed into history the old Eastern Liberal Republicans.
As for the Great Society in detail: the War on Poverty for instance was incredibly successful. By 1980, even after the rather nasty '70s, the rate was about half what it was in 1964—although most funding/resources had been in the '64-'66 period before the conservatives took back over the House (Republicans + Southern Democrats) and a lousy $70 million anti-rat program had trouble getting passed.
In other aspects… less successful. The failure to bring in national health care or health insurance and instead only the limited Medicare type stuff[1] would—combined with Nixon making them slaved to inflation programs—lead to entitlements becoming a huge factor in the US rarely balancing the budget and thus some the consequences they're suffering now: falling dollar & loss of investor confidence.
For larger efftects, without having to pay for Viet Nam clearly the Great Society will not impact the US government's finances as badly as it did IOTL, on the other hand it still has only two years before the conservatives win in 1966 so no Viet Nam means at best a touch larger Great Society in the first Johnson term.
A second Johnson term, as seems likely without Viet Nam, may mean some additional or expanded Great Society programs (not least because the 1968 Congressional Republicans did not hugely improve their position IOTL)
At best I can posit no more than a moderately larger Great Society. Given that Nixon offered a deal for national health insurance IOTL (and Edward Kennedy turned him down because he wanted national government run healthcare) it seems quite possible that Johnson could get through national health insurance. However that's probably as far as it goes.
In other news, no Nixon means China remains hostile but no Viet Nam means SEAsia has a lot less crazy (Poi Pot) leaders and way fewer dead people. No Nixon also means no entitlements, which means American social programs remain affordable if somewhat underfunded. National health insurance should mean lower US spending on healthcare as well as lifting some burden off US business which will both help in the long run.
No Viet Nam means the Draft still exists, with possible consequences down the road.
Two Johnson terms (assuming he lives through his second) means we almost certainly see a RFK-Reagan fight in 1972 (RFK would have plenty of establishment support even with VP Humphrey; President Humphrey may beat RFK however). Both hold strong appeal to "Reagan Democrats" with RFK getting poor/black/minority turn-out and Reagan winning college grads (as Republicans used to do) and the Midwest/South thus leaving blue collar voters the swing group. I suspect it hinges on a couple industrial states, a couple Southern states, Texas, and California.
Either could plausibly win, and it depends on their VPs, conditions of the time, and success of Johnson's second term.
A Reagan win in 1972 probably means military reform combined with a strong anti-Communist line; a Kennedy-Johnson style middle class tax cut (too early for supply-side economics), possibly reduced spending on social programs by a little (Social Security in particular).
Interestingly Reagan may attempt to mortally weaken Social Security by introducing a guaranteed annual income/negative income tax. Nixon tried to bring one in, but failed to introduce matching program cuts as conservatives wanted. A tax reform bill that gives everyone (perhaps on a means-tested scale) some money each year in return for weakening welfare & Social Security may be acceptable to enough conservatives in Congress. It would also be hard for Democrats to fight against it: yes, SS & welfare are being weakened—but you get money every year in return.
Another Reagan option is to stop bussing. Although this may bring a fair amount of segregation in the short term, the long term consequences for US cities are generally positive: less suburbs, higher density, larger and broader tax base, fewer ghettos, more jobs, and so on.
An RFK win in 1972 is not dissimilar in broad outlines. As with Reagan RFK would be throwing a lot of attention at the middle class/blue collar groups. However RFK would almost certainly spend more time on the poor & minorities.
Although he'd approach it from a very different angle than Reagan, an RFK Presidency would almost certainly spend a lot of time & money on cities, which is a very good thing in '70s USA.
No Viet Nam keeps the Democratic Party together; no Watergate keeps the Republican Party together. Conservative Bourbon Democrats of the South and the Eastern Establishment Republicans will play a larger role in their respective parties, acting as a moderating influence.
This should also keep the playing field wider—no Red/Blue division. Particularly the Democrats should be more competitive on a Presidential level 1968-1988 as well as holding on better in the South, while the Republicans will likely keep having a chance at California and the Northern states past 1992.
Note that most of these benefits flow from no Viet Nam and no Nixon, an expanded Great Society's only possible real benefit lies in keeping funding on programs that work (poverty) and with a bit of a stretch national health insurance.
[1] Note that on a per capita basis the US spends more than any other Western nation on healthcare, without the national system Europe/Canada has to show for it. Something like 16% of GDP versus ~10% in other countries.