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League of Nations vs UN
Well to start with, the league of nations was a European congregation of impotency. In more technical terms, it was bollocks.
The United Nations is this lobby room wherein nations with a lot of economic and military muscle pitch their cases and intimidate countries which are economically dependent on them.
Ok let’s get a little more serious.
The UN is not just an organization to prevent security situations. It aims at creating economic, social and cultural ties between member nations (or so it claims at least). It has international banking organizations under its wings, forums for international trade, child relief, humanitarian aid, and on top of this, a security council. As to whether any or all of this are of any real use is something that is highly debatable and this question doesn’t demand that. But all the same, on paper this is how it is.
The league of Nations was made as a primitive form of collective security at a time when Europe was not interested in collective security. Its members were in it for the perks of imposing their foreign policy and economic agenda on the defeated nations of WWI. Italy, Germany and Hungary pulled out when they got the chance. With no other nation present to uphold its offices and importance, it was let to disappear in history. The League of Nations was an example of how utterly a failure of a concept can collective security be. It exposed that collective security requires wholly, the relinquishing of individual nationhood, leading to a collective economic, social and military order - like a world government.
This more or less sums up the functional differences of the two organizations.