If the implicit assumption in the question is: why wasn't there a United Nations intervention in the Vietnam War like there was in the Korean War, the answer is that it's largely because the UN intervention in Korea was possible due to an accident of circumstances that was not to be repeated during the Cold War.

United Nations involvement in the Korean War was authorized by UN Security Council Resolution 82, which in its Section three "Calls upon all Member States to render every assistance to the United Nations in the execution of this resolution [ie, stopping North Korean aggression against South Korea] and to refrain from giving assistance to the North Korean authorities." A Security Council resolution can only pass if all of its permanent members (the US, UK, France, China, USSR/Russia) do not use their veto. How did the Soviets let UNSCR 82 pass?

Since January 1950, the USSR had been boycotting the Security Council in protest over the People's Republic of China not getting the Security Council seat held by the Republic of China (the PRC would get the seat in 1971). This turned out to be something of a tactical blunder, because the Soviets were otherwise infamous for vetoing Security Council resolutions - in the first ten years of the United Nations, they vetoed 79 resolutions, compared to two vetoes from France, one from China, and none from the US or UK. Andrei Gromyko, the then Soviet UN Ambassador from 1946 to 1948, personally vetoed some thirty or so resolutions and received the moniker "Mr. Nyet" in the Western press as a result.

Because of the boycott, the US was able to shepherd through UNSCR 82 without a threat of a Soviet veto. Gromyko, now Deputy Foreign Minister, complained of the "rubber-stamping" and "illegality" of the resolution, but frankly couldn't do anything about it at that point. From that point on, the Soviets learned that it was better to be present and saying "Nyet" than not be present at all.

Of course, it's worth separating the idea of peacemaking (intervening in conflicts) from peacekeeping (preserving or implementing cease-fire and peace deals). Technically, the United Nations is supposed to do the latter, not the former, although objectives can get very blurred.

At the time of the Vietnam War, there were very, very few precedents for UN peacekeeping missions. A UN list of past peacekeeping missions will show that the vast, vast majority of these operations occurred at the end of the Cold War or in the post Cold War world (ie, when the permanent members of the UN Security Council mostly agreed on allowing such missions to happen).

The one major exception to this was ONUC - the United Nations Operation in the Congo, which operated during the Congo Crisis from 1960 to 1964. This was authorized originally under UNSCR 143 (approved by the US and USSR, but France, China and the UK abstained from voting) which authorized UN support for assisting in the decolonization of Belgian Congo. After the assassination of Patrice Lumumba, the mandate was interpreted as one to prevent local rebellion and secession movements in Congo, notably in Katanga. Of course, this mission did not do anything to prevent American and Soviet intervention in the wider Congo crisis, nor did it prevent the widespread Simba revolt (which saw the Congolese government employ Belgian paratroopers and white mercenaries to combat). If that wasn't enough, UN Secretary-General Dag Hammarskjöld himself died in a plane crash in the Congo in 1961.

So in the 1960s, UN peacekeeping missions were still a very controversial option, that based off of its one major operation to date had a very limited track record. This, coupled with the fact that two of the permanent Security Council members (the US and USSR) were supporting opposite sides in the Vietnam conflict meant that any sort of peacekeeping mission in Vietnam wouldn't ever pass the Security Council.

And who would the UN have intervened on behalf of? The Security Council included the USSR which would have vetoed anything supporting the US, and also included the US which would have vetoed anything opposing itself.

UN Secretary-General U Thant from Myanmar was actually horrified as news of American-sponsored genocide dominated headlines worldwide, exacerbated by the geographical proximity between Myanmar and Vietnam. He mustered various organs of the UN to criticise American actions throughout his tenure, and worked to broker secret direct peace talks between Hanoi, Saigon and Washington. These attempts were rejected outright by the Johnson Administration so there wasn't too much he could do.