"Ich kann es nicht tun."
Translation:I cannot do it.
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That's not easy to explain. In many cases (like here) you can use them interchangeably. But in some, you can't.
Machen (to make) has the additional meaning of "to create", which tun (to do) hasn't:
ich mache Frühstück - I make breakfast
Then there are many combined words which always use the one or the other, which you just have to learn:
Ich mache die Tür auf (aufmachen) - I open the door You can't say "ich tue die Tür auf"
es tut weh (wehtun) - it hurts Only little children sometimes say "es macht weh"
Whoever taught you that was wrong. Anyone interested in this, have a look here:
I think Duolingo's website might pre-load one correct answer from the database, and only search the database when you answer something else, hence taking a bit longer sometimes.
Your sentence is grammatically fine - totally normal. No reason not to use it, although if you're telling someone you can't do something you might want to be less direct about it, depending on the situation!