In actual history, the Supreme Court ruled in 1971 in New York Times v. US that prior restraint is illegal, upholding its earlier decision in 1931 in Near v. Minnesota.
In my alternate history, the Supreme Court rules that the US does not have to prove that publication would cause irreparable harm to national security in order to prior restrain publication. This does not overturn Near v. Minnesota, but rather broadens the scope of its exceptions.
Nixon, still fearing a left-wing radical conspiracy to delegitimize his administration following the publication of the Pentagon Papers, sends a bill to Congress called the News Management Act. This calls for the creation of the Department of News Management (henceforth DNM), which would require all newspapers to submit their copies and get pre-approval from the government before publication. Likewise, news channels need to send scripts to the government before going to air. If they report without pre-approval, they risk being shut down. This gets approved in late 1973.1
Of course, the newspapers sue on the grounds of violation of the First Amendment. Shortly before the Watergate break-in, the Supreme Court rules that the DNM is legal, as it only prevents national security-related matters from being published; all other articles are still able to be published, and so freedom of the press is preserved.
The government being the government, this freedom is used for political gain, under the guise of national security. ("If that person gets elected, the whole country will be invaded by Commies!") And how can anyone complain - what are they going to do, print it in the papers? That would delegitimize the sitting President, leading to national security issues.
The first major change would be Nebraska Press Association v. Stuart in 1976, where, in actual history, the Supreme Court ruled that, since prior restraint is illegal, the media is free to print whatever they want about ongoing court cases without worrying about influencing the jury. Because in this alternate timeline prior restraint is perfectly legal for "national security," the government is free to influence cases by choosing which ones the news can report on.
As a corollary to the previous point, I don't think Chappaquiddick (which I'm aware happened before Nebraska) or Whitewater would have ended differently. Neither Nixon nor Bush 41 would feel it's worth censoring. Nixon probably would make sure that Chappaquiddick gets aired so that Ted Kennedy doesn't run in '72 and '76, as actually happened.
Really, I think history would overall remain the same (except for the public distrust of the media) until the late 2000's, with the advent of social media. Now people are getting their news online in the masses. The government doesn't like this loophole and passes the Internet News Management Act in 2007, extending the DNM's censorship to the Internet. This means that any news site or social media platform must be moderated by the government before anything is published.
As the Internet gets larger and larger, the government struggles to keep up. A new division of the DNM is opened to work on this problem specifically. Because of this new problem, AI research is advanced by several years, and in late 2012, the DNM begins to employ new algorithms to help with the monitoring. The entire Internet is monitored by early 2016. Websites operating inside the US are blocked from publishing certain content and are taken down if they find a way to bypass the filters. Ones operating outside the US are simply blocked from entering.
Because the Internet, television, and print media are all being censored, Snowden no longer has a way to send PRISM to the masses in 2013, and we remain in the dark about the NSA's spying. (He could post it on blogs or whatever, as it's not until 2016 that the entire Internet is moderated, but it wouldn't become as widespread.) I think the privacy vs. security debate would start eventually, probably with 2016's San Bernardino case.
The one thing I'm unclear about is the 2016 election. There's no more election meddling, and I strongly believe that, because of Obama and Clinton's relationship, the media would be heavily slanted to the left at that point. What I'm unclear about is how this impacts the election: does Clinton now get elected, or would those who'd vote for Trump anyway still land him the majority in the electoral college?
1In an earlier draft, I said that the DNM was established fairly quickly, in early 1972. At first I thought that Watergate would happen pretty much the same way; the only thing that changes is that Woodward and Bernstein are unable to publish their articles in the Post about it. The burglars still plead guilty to perjury under pressure, and the case is led toward Nixon, who eventually resigns. However, I'm not sure anymore how important the Post's articles were, and to avoid this issue, I'm now saying that since bureaucracy takes time, the DNM doesn't get established until later 1973 - after Watergate has already sufficiently broken that covering up the story makes him look worse. Nixon still resigns in 1974, succeeded by Ford.
My main question is: is this a plausible timeline resulting from the Supreme Court ruling that prior restraint is legal?
I'd especially like to hear the legal ramifications of this, as while I've done a bit of legal research, is it likely that the theoretical 1972 lawsuit regarding the legality of the DNM would be ruled in favor of the government, given that the Pentagon Papers ruling was also ruled in favor of the government?
Some of you in the comments theorized that the Presidents elected in the interim may not have been elected in such a world. I'll have to look into their respective platforms, and especially their relationships with the candidates of the succeeding elections, but at the moment I feel secure in saying that, besides the 2016 election, the Presidential roster remains the same.