The big question here isn't so much how to transport Cthulhu into the 60s but what aspects of the Cthulhu mythos you want to capture and bring ahead. I think people have already explained the turmoil of the time that would seem to make it apt for this period, so instead I'm going to focus on Lovecraft's themes and how you could transport them ahead 30-40 years. (note: theme taken from the Lovecraftian Horror Wikipedia page)
Anti-anthrocentrism: Lovecraft liked to portray humans as insignificant beings in the universe, and the Space Age is a really great place to make this idea come home to roost, so to speak. It's also a time when humanity is really beginning to wrap up the mapping of the globe in a geographic sense, so there may be some final outcrop or two where your party could experience a new Colour Out of Space.
Additionally, this was a period of time when a lot of African and South American nations began to shrug off the yoke of colonialism and give self-rule a try. I don't want to debate the merits of colonialism or whether or not there's a "right" way to get one's nose out of another country's business, but historically speaking, many of the nation-states that emerged from these colonies treated their people brutally. This would be a neat place to run an adventure or campaign wherein, let's say, an obscure Congolese culture, driven to the breaking point of oppression by its new leaders, returns to the Old Ways of worship and calls upon [insert Cthulhu being of historical might here].
Preoccupation with viscerate texture: Lovecraft liked slimy and gross things. I don't have a lot to say about this for the 60s, other than that there was just as much of it then as there was in the 30s. Well, I guess one thing: among so many other things, the 60s saw the birth of environmentalism in its modern form, in amongst some of the worst industrial abuses out there. Take Minimata disease, for instance. Or the Love Canal, which was sold to Niagra Falls in the 60s but wasn't discovered to be a cesspool of industrial waste until the late 70s. Surely there is plenty of monster-fodder here.
Antiquarian writing style: This is more of a stylistic thing than something I think that you'd absolutely need to include in your alternate universe, except for flavor.
Detachment: This may seem off to the modern reader, as we have lived so long with the idea that you are your own person, but the 50s and 60s were both a time of close-knittedness and the beginnings of rebellion against being like everyone else. If you the reader were transplanted into the 50s and 60s today, the first thing you'd probably notice was the extremely high level of conformity, particularly in dress but in a lot of other things as well. Wearing blue jeans meant you were a rebel, not that you liked comfortable pants. All of these silly cliches in movies like Footloose (in which dancing was an act of rebellion in a small town) or Dead Poet's Society arise from this period.
At the same time, Stephen King has noted that there was a lot more general "weirdness" then than there is now, because the world has gotten bigger and some of the things that kept people from fitting in have now been solved by technology or our general rise in living quality. I think one of his examples was that you just don't see kids at school wearing the same shirt for 3 months straight, or kids with cleft palates, that sort of thing.
I think there are several ways you can use this. A town out in the middle of nowhere in the 60s might still be nearly as insular as during Lovecraft's time, in a way you just don't experience now. Heck, David Koresh still existed less than 20 years ago.
Helplessness and Hopelessness: This is a tough theme to mix into any RPG campaign, to be perfectly honest. That being said, the challenges of using this in the 60s isn't any greater, I don't think, than using it in the 30s. Maybe you could implant some false sense of hope on the part of some NPCs, like a general who wants to harness the poweer of Nyarlathahotep to defeat the commies or something similar. As the song goes, the only way to get to nothing is by having something in the first place.
Unanswered questions: Again, no more incompatible with the 60s than the 30s. There are, in fact, some "unanswered" questions from this period of time which you could have some fun "unravelling". For instance, the JFK assassination; in an alternate universe, maybe there really was a conspiracy and it wasn't, in fact, just some loony communist with dreams of grandeur. Maybe there's something to the Bermuda Triangle. Or Bigfoot (if memory served, the Patterson/Gimlin film surfaced in the late 60s). Leonard Nimoy came out with a (very silly) series about this kind of thing that was called In Search Of. This was the early 70s, not the 60s, but the culture hadn't moved forward that much.
Sanity's fragility and vulnerability: Among many other things, this period of time was when we saw an explosion of attention given both to psychology as a whole and to the mentally ill. One Flew Over the Cuckoo's Nest was released as a novel in 1962, for instance. MK-ULTRA, which is probably worth an entire Cthulhu-esque campaign in and of itself, was in full force during this period of time. There is, in my opinion, probably more fertile ground here for adventure ideas than in the 30s, where the insane were often locked up in asylums and largely forgotten about (again, still happened in the 60s but this practice was really beginning to worry people) or locked up in the attics of wealthier families.
Questionable parentage: Um... no comment.
So yeah, I think there's a great deal of fertile ground here. Getting the feel of the time is probably more important than making sure the characters are there to witness the Mississippi Burning incident, the debut of the Beatles on Ed Sullivan, and/or the Summer of Love. I wish I had books to recommend but off-hand all I can think of is David Halberstam's The Fifties, which, as the name implies, is a little too early.
Best Answer
This interview with Robin D. Laws might clarify some of the differences.
Basically, your investigators won't be denied essential clues throughout the course of the game. GUMSHOE prevents the awkward "Oh, you really missed that investigation roll. Your seasoned detective manages to completely miss any clues in the well-stocked library with a post-it note on one of the books that says 'READ ME'".
I really like the flavor of the Trail of Cthulhu book. Instead of giving you how many hit points Yog-Sothoth might have, it gives you several scenarios for what Yog-Sothoth might be, and what it's intentions may be. So even players intimately familiar with the Mythos may not know what is really going on.
Overall, it's a refreshing take on Call of Cthulhu, and is definitely worth checking out, even if you don't decide to use the mechanics.