I guess it happened because of the US involvement.
When forming the League of Nations in the late 1910’s nobody could imagine such a thing as ‘not involving’ the US. But once the League was formed, the US was NOT included, and the other countries did not have enough power to control everyone. Like when Japan was reproved for its actions in the Pacific, they just a unassigned league of Nations and nobody could stop the process. Also, the important thing to mention is that the founder of the league of nations, Wilson, wasn’t really participating, that is why nobody was aware of what is going on or what they are doing. There were also issues like blocked unpopular decisions or public votings. The organization of the process always plays the key role, that’s what we have to learn from history.
Still, with the involvement of United States in the United Nations, it went better, for the US is one of the biggest powers in the world. Meaning both, military and financial aspects. The US is a member of the Security Council and also is a world's superpower, so it can easily cause a lot of damage to those, who won’t follow the rules. That is why United Nations was way more successful in their activity than League of Nations, but we cannot state their ‘full success”, of course. Any country at any moment can skip participation and cooperation, even by being punished by different kinds of embargos and charges, or, at the very end, by invasions, and United Nations won’t be able to stop it all.
The bright example of it can be the Korean War. Technically, United Nations were involved, the United States was delivering something about 90% of all the used resources, while also having the full command of the process and holding the lead for all the operations. And what was the result of it all?
To cut a long story short, I can say that UN was more successful because they included not only IS as a significant power, but also many other powerful countries at that time. Everyone was investing in the success of the United Nations after World War II. That was the problem that League of Nations faced at the World War I and couldn’t resolve.
The United Nations succeeded because it included all of the major world powers from the start and because everyone knew of the potentialities of another World War if there was another breakdown in international relations. Each authority was thoroughly and personally invested in the success of the United Nations after the World War II, in a way that was lacking for the League of Nations after the World War I.