In this case, President Nixon was ordered to hand over the tapes that ensured that he would have to resign as president. Thus, it clearly was a defeat for him. However, the case was a partial victory for the Presidency as an institution. It affirmed the presence of a strong executive privilege that applied in many cases (though not in the case of Nixon’s tapes).
Up until this case, the Supreme Court had never ruled on the idea that presidents had the right to refuse to cooperate with the other branches of government on the basis of executive privilege. In this case, however, it explicitly asserted that the Presidency did hold such a power, regardless of the fact that it is not clearly given by the Constitution.
What this meant is that the Presidency gained a power that it had not clearly had before. It gained the power to reject demands from the other branches of government in cases of national security or other areas in which secrecy was necessary. The Court merely said that Nixon’s case did not appear to fall into either of these categories.
Thus, while President Nixon lost the case, it can be argued that the Presidency won the case because it now had an explicit executive privilege.