"An old worker's story" sounds like a story that is very likely to be told BY an old worker, or a story a lot of old workers would tell. Here's an example---> Mr. Tom sat down on the stairs and to tell me an old worker's story. "One day long ago," he said. "back when I worked at the meat packaging plant..." A story of an old worker sounds like the story is ABOUT an old worker. Like, if I told you a story about my grandfather working at a milk factory.

They could have the same meaning, but the first one probably refers to a story told by an old worker, where the second one might be more likely to refer to a story about an old worker.