D&D is an RPG. A role playing game. Characters play a role in a story. That is the core of the game, as directly stated in the name, and as described in the core books. The first line of the introduction of the PHB says it: "[D&D] is about storytelling..."
Adventuring is a large part of many D&D stories, but these characters have lives outside the dungeon. You can ignore those lives, and focus on their tales only during their time with weapons in hand, but the game is designed to cover more than just the combat pillar. There are the exploration and social pillar experiences in the game, and there are a lot of rules that support those elements. Money can play a huge part of the social pillar (especially in a political context), and also a large role in the exploration pillar. A party with no money that finds a ruin and decides to explore it does so by themselves, but if they have the money, they might have a base camp set up, or hirelings to perform tasks for them.
If you find the money doesn't make any difference in your story, my advice is to consider what stories could you add to your tales that do revolve around money. As there are so, so, so, so, so many options, there is likely to be something to capitalize upon that can make your game even more fun. In my experience, when money doesn't matter, it is because the DM and players are not collectively utilizing the full breadth of the game options available to them, and are missing out on fun elements.