In American English (perhaps British as well), "housework" and "homework" have different meanings. "Housework" refers to working around the house, such as dusting, vacuuming, etc., whereas "homework" refers to doing some school assignments at home, after school is out.
Homework can be completed outside the home, such as in a library. In order to do housework, however, one must be at one's house. How and when did two words which seem so similar at first glance gain such different meanings?
According to the OED, the original meaning of "homework" does conflate much more obviously with "housework," with the former being defined, above all, as:
Work done at home, esp. as distinguished from work done in a shop or factory.
The earliest citation is a hearty piece of precious advice from a sermon from the 1680s:
Wherefore let every Man, in the first place, look after his Homework; what he hath to do at Home.
Less vague examples of homework were given in later quotes: Spinning, quilting, and embroidery. This crafty and practical usage seems, however, to be an obsolescent meaning of homework, with the last use from the '30s. But the word "home worker" (doing low-paying piecework) lives on, preserving this original meaning of "homework":
Most home workers are women. They need the flexibility of working hours that home work allows. (Guardian, 1973)
The second—and now primary—meaning of "review/preparatory school work despised by youth" didn't appear until much later (late 19th century), but it's thriving and strong, having quickly overtaken the original meaning.
The first citation of "house-work" from the OED (which hyphenates it) is from mid-19th century. Its meaning has always been as it is now: "the work done to keep a house orderly (and housewares clean)," diligently by housewives and begrudgingly by house-servants:
While the boys are engaged in out-door work, the girls could be employed in sewing or house-work. (Report of the Commissioner of Indian Affairs, 1871)
Here it contrasts homework (sewing) with housework [other activities].
As FumbleFingers's Ngram shows, it has also been used in texts in its unhyphenated form, which the OED has chosen not to include, as Alenanno first noted above in the comments, although it does feature a sub-entry for the spaced "house work" (definition-less, with a single late-19th-century quote).
So the two words would seem to have diverged after the first (homework) took on a specialized meaning relatively late in its life. Now let's find the courage to get back to doing either/both!
I think it's pretty clear housework came first, as shown by this. Apart from anything else, the activity now called homework probably didn't actually occur very often in a world where many children either didn't go to school at all, or had limited access to books, writing materials, etc. at home.
But it's worth mentioning that some (non-working) women who don't really like to call themselves (or be called) a 'housewife' will accept 'home-maker'. There are various subtle differences between house and home, but they're pretty much the same thing in most contexts, notwithstanding the 1964 song A House Is Not a Home.
