Other than underlying intent -- providing an international forum for the peaceful resolution of disputes between nations -- the differences between the League of Nations and the United Nations outweigh the similarities.
Largely the vision of U.S. President Woodrow Wilson, who included the creation of a League of Nations as part of his famous "14 Points" intended to prevent another conflagation on the scale of the Great War, Wilson failed in his efforts at having the League's establishment included in the Treaty of Versaille. He was, however, more successful in attaining agreement among the victorious nations to have it established as part of a final framework agreement.
The Covenant of the League of Nations was a lengthy document that recognized the right of all nations to independence and security from outside aggression. The U.S. Senate, however, declined to ratify the treaty establishing the League of Nations because of the majority's opposition to the mandatory consultative process the treaty established, which some Americans feared would result in the United States' being pulled into overseas conflicts to which it otherwise would have no part. Consequently, the United States, which conceptualized and fought for creation of the League of Nations never became a member.
The post-World War I world was fraught with economic and political instability, with the onset of the Great Depression and the rise of the National Socialist Party (the Nazis) in Germany unravelling British and French aspirations for Europe. In Russia, revolutions and civil war marginalized that country's role in establishing a new arrangement to prevent another major war. The League of Nations would prove a well-intentioned footnote to history, but nothing more.
As World War II raged, President Franklin Roosevelt and the other leaders of the anti-Axis alliance began to envision a new international framework by which to ensure the Axis' defeat and the creation of a structure to prevent the outbreak of another world war. As the war drew to a close, 50 countries assembled in San Francisco at eh United Nations Conference on International Organization and the result was the United Nations Charter.
Like the League of Nations, the U.N. Charter set as its foundation the inviolability of international borders against foreign aggression. Also similar to the League, the new organization would exist to provide a forum for nations to debate and negotiate a resolution to their differences. Far more than the League, the United Nations would grow to become a massive, sometimes unwieldy bureaucracy with numerous subordinate agencies focusing on areas like human rights, child welfare, health, economic development, and the promotion of political freedom.
The main difference between the two organizations is the center of power within the United Nations. As stated, during the years of the League of Nations, the same countries that repeatedly fell into major conflicts again dominated international discussions. By the end of World War II and the creation of the U.N., there were two new superpowers, both with permanent seats on the U.N. Security Council and both holding veto power over binding resolutions. That arrangement would effectively hamstring the U.N.'s effectiveness as a forum for conflict resolution.