You are suspected of a crime and have to tell the police exactly what you did on a certain night. You tell them:
I drove directly home after leaving work.
This is a more formal expression for explaining the idea of "going straight to somewhere". It is used when you need to describe exactly what you did.
Many different action verbs can be used with this phrase: drive, walk, go, run, fly, etc.
This expression is also formal and exact. Saying that you did something "after work" could really mean any time after your work hours. "After leaving work" specifically means the next thing soon after leaving.
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