The 1960s were a time of spiritual exploration and social awakening. In the world of the Cthulhu mythos, these things could represent the sort of knowledge that leads to understanding things that should never be learned. Inside every commune lurks a dangerous cult (artists and sensitives were particularly susceptible to Cthulhu's dreams); "free love" is a mask for Shub-Niggurath and Y'Golonac; and Transcendental Meditation opens the mind to the Dreamlands and the mad piping of Azathoth.
(I have to, at this point, allude to Lovecraft's racially problematic points of view, and suggest that it might be fruitful to contrast them with the advances gained by the Civil Rights movement. That's a can of worms, though, and I'll leave the implementation of such things as an exercise for the reader.)
As for the advances in technology, they seem to come in two flavors: incremental (better guns, faster and more reliable cars) and innovative (space travel, advances in computing and materials technology). The first is unlikely to have a significant impact: Chaosium's Cthulhu Now! demonstrates that you can run a fairly traditional CoC game in a modern (or at least later-20th-century milieu) without too much change. The great innovations of the age are largely kept in the upper echelons of political and military power; for more on that, take a look at Pagan's Delta Green setting.