From Sanford Ungar's The Papers & The Papers: An Account of the Legal and Political Battle over The Pentagon Papers (1972):
As months went by, legal scholars and journalists alike began to realize that the 6-3 Supreme Court decision in the cases of New York Times v. United States and United States v. Washington Post had been a rather hollow victory for the press. The unsigned opinion of the court said only that the government had failed to carry its heavy burden of proof to obtain a prior restrain, and the individual justices' opinions had as many nasty words for the press as for the government. Many editors felt strongly that the newspapers were worse off after the Pentagon Papers than they had been before it; and in private conversations high Justice Department officials concurred in that assessment. "We proved on thing emphatically," said on Justice policy maker, "that there can be prior restrain of publication while a case is being reviewed in the courts." (303-304)
I think this is part of the explanation of why the New York Times institutionally seems compelled to go to the government.