Sie hat den Hund nicht angenommen. - they offered the dog to her but she refused it.
Sie hat den Hund nicht akzeptiert. could mean that as well, but could also mean that she refused it mentally (if that makes sense): that she thinks that the dog is somehow reprehensible. Or that she didn't "take him into her heart". For example, you might talk about a child not "accepting" their father's new wife -- that's not really got anything to do with an offer that gets accepted or refused but is a more abstract kind of "acceptance".
And Sie hat den Hund nicht angenommen could perhaps also mean that she did not adopt it.
If I'm not mistaken, the latter would be
"Sie akzeptierte den Hund nicht." but that form is used mainly in a narrative past. Spoken German would be what Duolingo taught us.
This article can help you more https://www.thoughtco.com/g00/german-past-tenses-how-to-use-4069394
Edited: Thanks @mizinamo.
Hello, everyone! To assume that Perfekt and Present Perfect are the same is a huge mistake. Every German teacher as a foreign language knows this. The distinction between those verbs tenses is not made clear in this section. You should probably complement the sentences with some expressions of time to make the difference more obvious. Best regards, Filo