What has prevented the United Nations from failing the same way as the League of Nations?

The role of the United Nations changed significantly. The League of Nations was a rather shoddy attempt, that honestly only had one working goal: war reparations. It failed because it both didn't have the backing of nations interested in aiding it, and because it was a fairly lost cause to begin with.

The United Nations, though, has existed because 1) it serves numerous purposes, and 2) because it established a very important mantle during the Cold War: the open facilitation of conversation on neutral ground. In terms of the first, it has essentially served as a facilitator for numerous activities and goals, from economic aid to facilitation of services from awareness of human rights violations to encouraging collaboration on international projects.

Some examples of this, examples that a lot of people who aren't aware of the United Nations typically don't know about:

World Bank: Offers advice and facilitates aid from developed countries to many developing countries, establishing not only the financial means for them to undergo significant infrastructure development projects, but the finance-driven incentives to make these projects helpful and useful.

WIPO (World Intellectual Property Organization): significant contributions to facilitation of copyright information, particularly in pharmaceuticals and brand names, which heavily encourages companies in westernized nations to make their products available to nations that would otherwise feel free to steal it without repercussions under the rule of national sovereignty.

UN ECOSOC - Essentially serves as a forum for numerous intergovernmental bodies, facilitating discussion and collaboration across multiple groups. The UN general assembly allows governments to communicate; ECOSOC allows organizations to communicate. This has had significant impacts in the formation of bodies such as the Intergovernmental Panel on Climate Change, which houses the single largest body of climate change data and expertise in the entire world.

That isn't to say that the UN is perfect; the Security Council is functionally little more than a means to politely say "do not do this or we will nuke you." The ICC's limited jurisdiction given national sovereignty and very lengthy processes have a very difficult time compensating for the immense manhours and resources devoted to it. But it has so many functions on so many levels that it more than makes up for its failures or weak points, at least in comparison to the League of Nations, which was fairly singularly focused and oriented around European and colonial affairs.

There could be multiple perspectives to looking at this, but in my opinion its the following three factors:

1. Economic Progress and Globalization: Since WW2, most countries have been so busy in rebuilding their Economies or sustaining Growth and Development, that war never got a priority that it once used to get (like during the 1800-1900 era). Especially since the 1990s when Globalization and Liberalization has taken speed, any country will have a lot more to loose (in terms of market establishment and decline in growth) by going to war (particularly for the super-powers who actually have that potential).
2. Nuclear Deterrent: America's demonstration of almost instant annihilation by dropping atom bombs at Japan in 1945 has since acted as a major deterrent for any other country to start a war-like scenario. For the first time, people knew that such great destruction could be caused within seconds of timescale that entire cities could be annihilated. The result: Each country started racing towards building more and more sophisticated nuclear weapons (America's total defense budget is more than a small country's entire economy's worth!). But no super-power will ever use it today, because due to sophisticated radar technologies, the enemy country will know even before the weapon has entered their borders, and their nuclear weapons would be instantly triggered causing absolute annihilation to both sides.
3. The United States: Since WW2, the USA has emerged as the single largest super-power in the world. Since it demonstrated its power by dropping the atom bombs in Japan, USA has since acted as a war inhibitor by keeping other smaller countries in check. USA has had its way wherever it wanted, and this has discouraged many other countries from going to war. For instance, think about the Iraq invasion of Kuwait in 1991. What would have happened if USA hadn't intervened by means of Gulf War? Iraq would have turned into a bully by invading more Kuwaits and the whole of middle-east would have plunged into a world-war! (Its a different matter that USA had its own interests in that war - OIL, but the result has been an aversion of a major war.)

To summarize, its not the UN per se which has kept the countries from going to a world war on its own, but these other important factors that have acted as major war deterrents, have also kept the UN from falling apart.

EDIT: In response to doubts raised by @SJuan76, here is the link describing how the USA, by means of nuclear and extended deterrents, just talked their opponents into stopping any war plans. To be clear, I'm not implying here that USA has been a peace-keeper because of altruism and benevolence. Like all countries, America is also looking after its own interests, but a side-effect of that is that many smaller countries have thought twice before plunging into a war due to fear of inviting the bigger monster's wrath (think again Iraq-Kuwait scenario and Gulf war).

As regards Indo-Pak conflicts, that has been going on since the 1947 partition of India and Pakistan. The US might have tried to take advantage or sides in this ongoing proxy-war, but they are by no means, the causal factor.

First, I think the United Nations is a barely functioning body, diplomatically. It serves many worthy causes in some of its agencies, such as the World Health Organization and the World Food Program, but for its main stated purpose it isn't much of a factor.

However, the UN is sort of an arbiter of last resort for the smaller powers, because its decisions are backed by the larger powers.

As for the greater powers, generally those with permanent seats on the security council, the name of the game is Mutually Assured Destruction. Once nuclear weapons became common, war between the major powers is no longer winnable, and is therefore avoided.