The females of these species are very different from each other as seen below. Take note how the House Finch is uniform brown lacking the light stripes over the eye brows (supercillium) and from the corner of the bill. The female Purple Finch and immature birds are also more blotchy spotted on the breast and sides while the House Finch has much thinner stripes.
Both adult males have similar coloration but with differences that even I for a long time had a hard time recognizing. What I ended up learning from some excellent long time birders is to check out the field marks because there are obvious differences between them. Although there will be that one bird that shows up that I might question, I do have a far easier time with them.
With the adult males, the Purple Finches tend to be more of a berry color while the House Finches are much more reddish. There is obvious streaking on the sides of the House Finches lacking in Purple Finches. Over the backs of the Purples will be that same berry coloring as well as a wash of it on the upper wing coverts. Wings of House Finches are brown.
About Birding http://birding.about.com/od/identifyingbirds/a/House-Finch-Or-Purple-Finch.htm
South Dakota Birds and Birding http://www.sdakotabirds.com/diffids/house_purple.htm
Slideshow to images of House Finches
Slideshow to images of Purple Finches