If you're launching Reaper via the Dock icon, it might still be linked to the older version. Therefore, try if removing the Reaper icon from the Dock and adding it again helps.
If this doesn't solve the problem: what version of Reaper is currently installed that you're trying to update?
General side note: if you're updating from a version prior to 6.29, the naming of the .app file got changed:
https://www.youtube.com/watch?v=5ms6rgQobNA&t=364s