Bill doesn’t run or swim.
Bill doesn’t run and doesn’t swim.
What is the difference in meaning between these two sentences?
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In your particular case, they are the same. however normally you wouldn't use 'and' in this case, as it's a more complex sentence structure.
How 'and' would normally be used, would be as both happening together. Like "John doesn't drink and drive." This doesn't mean he doesn't do either in their own, just that he doesn't drink while driving (or shortly before would be implied).
Bill doesn't run or swim = Bill doesn't run. Bill doesn't swim.
Bill doesn't run and swim. ---> Bill doesn't run and swim at the same time. (It's not possible too.)
Bill doesn't run, and he doesn't swim, either. (Bill doesn't run and doesn't swim.) = Bill doesn't run. Bill doesn't swim.
But this sentence is long and cumbersome, so the shorter negative sentences with 'or' are preferable.