It is less vivid; "raced after" emits a feeling of either action, suspense, or energy. It allows the reader to connect and fully understand what is going on. "ran after" is vague and emits little high energy.
Answer:
The primary theme in An Occurrence at Owl Creek Bridge is (C) the line between dreams and reality is thin. Throughout the course of the short story, the illusion of Farquhar represents a reality for the readers.