Translation:E kākau maiau.
I agree with your comparison except that the DL key doesn't include the 'ami "i". If one is correct then they both should be correct. They both fit this pattern: E- adjective - subject.
As you said, Duolingo already accept both of these for Do it neatly. E maiau ka hana. E hana maiau.
But the DL key doesn't have the same symmetry for Write (it) neatly.
E maiau ke kakau. = Be neat in writing.
E kakau maiau. = Write neatly. (also could be Write it neatly.)
Different ways to say the same thing.