I'm not entirely sure I understand your specific questions. Perhaps a few general comments will help. Depending on context they can both have very similar meanings.
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poppymanI'm tired of doing my homework" ... "I'm done with my homework"Two completely different meanings. Neither through with nor done with indicates in any direct way anything about being tired. If you're tired of something, you have to say you're tired of it. through with won't work, and done with won't work either.
poppymanI thought I could use that instead of "sick of"No. You can use these:
poppyman"I'm done with" can also "imply disgust or the desire to have n