Conservative is simply a policy of conservation, because of this we have to ask what was being "conserved" in the 1970s to draw ire towards Larry Flynt and what changed in the United States between the first publications of Playboy and Hustler.
Conservative A is not Conservative B.
What often gets lumped together as "conservative" is really a mishmash of ideologies, heuristics and social mores that have no one unifying factor. Because of this, it's not fair to call the 1950s a more "conservative" time than the 1970s, so much as specific affects of the 1950s are how we code conservatism today.
Playboy was a gentleman magazine for the sophisticated males interests, from all things culture to sex. In a time with defined (strict) gender roles. It was common practice from men to be members of fraternal organizations, or at least socialize away from their families in a pretentious symposium that today we'd call hanging out by any other name.
There is the long sighted gag of "I only read it for the articles," but for decades, the magazine did have compelling, general interests articles. Three time Pulizer winner Norman Mailer, sci-fi legend Ray Bradbury, Roald Dahl, Margret Atwood, and Kurt Vonnegut Jr make up some of the most prestigious names to write for Playboy during its height. The magazine held interviews with politicians, athletes and movie stars. While some people were most assuredly buying it for the pornographic images, that aspect of the magazine was not the main focus in its publishing (at first).
Playboy, despite its reputation, was never a pure pornographic magazine, so much as a men's magazine that featured pornographic imagery; even then it is debatable if the intent was pornographic or erotica. One of the distinctions (proposed by William Gehrke) is that erotica "seeks to tell a story." Playboy showcased women in a way that suggested a narrative fantasy. There props used in early photo shoots, or drawn into the more explicit cartoons, would include tobacco pipes, towels reading "his," armchairs etc... The model wasn't just some woman bearing it all for men, but a "gentleman's woman" portraying an idealized lifestyle that a reader would want to emulate. The big den, the nice furniture and, most prominently, a beautiful woman.
Fast forward to the 1970s and Larry Flynt's Hustler.
Hustler was, by all rights, a magazine created to distribute pornographic images. Through the foundation set by Playboy and the social revolutions of the 1960's, Flynt had a market that might not have been as receptive 15 years earlier, however (with no small irony), these same factors would foster a conservatism much less receptive to Flynts magazine.
The biggest change between the 50s and 70s was the 60s. The decade saw social change, sexual change and the birth of mainstream feminist philosophy.
Feminism brought with it a sexual revolution and a dramatic upheaval in the social structure of the United States. Equality in marriage (while obviously not overnight, nor universal) dramatically reshaped how men met on the weekends. It's around this time we see a decline in fraternal lodges and the Baby Boomer generation beginning to define the mores of the day. One of the things that changed with this was the advent of Christian Conservatism. The so called "silent majority" and later the "moral majority" would elect Nixon and Reagan to maintain the conservative values that got, as they felt, thrown out with the bathwater during the dynamic social changes in the decade prior.
Conservatism changed.
For some people they wanted to undo the changes of the previous decade, for others they wanted to only wanted to change parts of it, for many more they just wanted the relative progressivism to hold off for a while. One of the biggest changes to American conservatism was the shift to fiscal conservatism as the main point of focus with social conservatism being a secondary strategy (here's a link containing a 42 minute interview on this subject from 1981 (unfortunately the site it's on is rife with political punditry, but it's the first source I found when looking for the unedited interview)) built around social conservatism with an unspoken emphasis on undoing the changes of the 1960s.
What does all this have to do with Larry Flynt? Well, if there's one thing social conservatives and feminist at the time could agree on, it was porn. Gloria Steinem famously said "Erotica as different from pornography as love is from rape, as dignity is from humiliation, as partnership is fromp slavery, as pleasure is from pain." While later feminist writers have taken a much more nuanced approach (and today there is still debate in philosophical circles on pornography and feminism), the popular, "liberal" narrative was that pornography was inherently degrading. While, arguably, Hustler could not/would not exist without women's liberation and the sexual revolution that came with it, it's execution was unfavorable to conservative ideals within the feminist movement.
Christian Conservatives were never big fans of erotica/porn/Playboy, and the overt sexuality of Hustler was, by their eyes, sinful. With hustler being a result of the sexual revolution the decade prior, the magazine was veritable proof of concept to a slippery slope of moral depravity. The idea that sexual liberation breaks up families was a popular refrain (no less compounded when no fault divorce first adopted in California in1969). The family sexual dynamic was changing and to conservative christian groups, pornographic images that tempted men to lust after women who weren't their wives were a causal factor in this breakdown of family values.
In short, while Playboy was published to conservative values of the 1950s, Hustler was decidedly an affront to the conservative values of the 1970s. Add to the fact that pornography was, at the time, largely scorned by the talking heads of the feminist movement, and we get a decisive cultural backlash. Flynts list of political allies was small enough to take his publication to court.