Among the most obvious examples of the cultural impact of "No Star Trek" would be the removal of a the idea of a "optimistic, multi-cultural, peaceful future for humanity" as well as the elimination of several key influences from both the show and from actors William Shatner, Leonard Nimoy, etc.
For example, the first inter-racial kiss doesn't occur until 1977, in a scene where Leslie Uggams's character "Kizzy" is raped by Chuck Connors in the TV mini-series "Roots". The scene is doubly shocking, both for the sexual assault and for it being the first inter-racial kiss. Obviously no link to the "Plato's Stepchildren" episode kiss of Shatner and Nichelle Nichols exists. Therefore the first "inter-racial kiss on TV" is in the context of a rape.
Also, there is the mock-up Space Shuttle. Originally in both OTL and the althist, to be named the "Constitution". In OTL, it is renamed the "Enterprise" after a write-in campaign by Star Trek fans.
Notable as well is the absence of the career of William Shatner. Though lampooned, his characterization of "Kirk" is both an indelible part of Americana and an icon of "traditional male heroism". There is no similar "hero" in the "Lost In Space" saga (which superceds "Star Trek" in the althist) and "Luke Starkiller" ("Luke Skywalker" in the althist) is regarded rather dismissively (as in our timeline) over the more heroic and humorous "Han Solo" character.