The major conflict in "Contents of the Dead Man's Pocket" is between pursuing one's ambitions to get ahead in life and enjoying life in the moment. As the story opens, Tom has been sacrificing current pleasure for future gain. He has been spending evenings and weekends working on a project he hopes will get him to the top of the corporate ladder at work.
The story is a critique of the 1950s "rat race" and the materialism that set in after the end of the Great Depression and World War II. Tom is a symbol of that mindset, a well-meaning and earnest young man who puts material success ahead of all else.
More specifically, the tale focuses on the epiphany or flash of realization Tom has when it occurs to him, while standing on the ledge eleven stories above a New York City street, that he really might die at that moment. He understands that, should he fall to his death in the street below, his life will have been meaningless, wasted on work. This leads to Tom's revelation and regret of risking his life all for a page of research data and a potential promotion.
When he is able to reenter his apartment, Tom has a new view of life. In the conflict between deferred gratification and enjoying the moment, Tom has realized he needs to appreciate and spend time on more meaningful and fulfilling activities, such as going to the movies with his wife.