When I'm not working, I walk to the park.
When I don't work, I walk to the park.
Are the present continuous and present simple forms interchangeable in this sentence? If not, are there any nuances of meaning?
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not working is immediately understandable in the context you have supplied; whereas don't work requires some context to justify the simple present.
Your job might require you to work Monday, Wednesday, and Friday. You could say
I don't work on Tuesdays and Thursdays. And on the days I don't work, I go to the park.
The natural inclination of a native speaker, in this work context, would be to understand the simple present as a statement about your normal practice, habit, or schedule.
Now let's say your job requires you to work every Saturday, but this coming Saturday the business will be closed for some reason. You might phone a friend and suggest an outing:
I'm not working this Saturday. Do you want to drive down to the beach for the day?
You could say "I don't work this Saturday" in that particular context, but the simple present and "this" together create a little semantic dissonance, since "this" is particular and the simple present, in this context, is usually associated with the general, with normal behavior or practice. It would be more idiomatic, in this context, to say "I'm not working this Saturday."
To sum things up:
not working = not doing work for whatever reason
don't work = not working per some schedule
There are many other nuances; this is just an attempt to focus on the primary difference between the two