In English “housework” and “homework” are distinguished by the first part of each compound being different “house” + “work” versus “home” + “work.” In German the same two words are distinguished by the second part of the compound: die Hausarbeit and die Hausaufgabe. Breaking them apart we get das Haus + die Arbeit – which can be translated as “house” + “work” – and das Haus + die Aufgabe – which can be translated as “house” + “task.”
“Housework” might be considered “work done to the house or for the house” and “homework” might be considered “work done at home” and, in particular, schoolwork done at home. We could describe work done to or for one’s house as “work done to or for one’s home,” but we don’t say “work done *at house” in English, the collocation with “at” and without an article is “at home” (although “at the house” is possible). In pondering this, it occurred to me that to say “at home” in German, you say typically say zu Hause, although you can say zu Haus. This Hause form is used regularly with two other prepositions im and nach: for example, im Hause bleiben – “to stay indoors” and auf dem Weg nach Hause – “on the way home.” Dict.cc and Duden also include the word das Zuhause for which dict.cc offers the translations “home” and “crib” (in the slang sense of this term rather than a type of bed for a child!) and further searching uncovered the idiomatic phrase wie ein zweites Zuhause – “like a home away from home” – and this article about bookstores in Hamburg!
And I can’t resist throwing in another compound with das Haus – der Hausarzt/die Hausärztin – the “family doctor” or “general practitioner” and not the doctor who works from home, nor the doctor for a particular house, nor someone who repairs houses, although perhaps this is the person you might consult if you’ve lost the energy to complete your housework or homework!