Here's what the curmudgeonly and thoroughly wonderful Oxford American Writer's Thesaurus, which I recommend heartily to anyone in the thesaurus market, has to offer on the tricksy subject of that v. which--
"There is a widespread ignorance about how to use that as a relative pronoun, and two common that-errors are so severe that teachers, editors, and other high-end readers will make unkind judgments about you if you commit them. The first is to use which when you need that. Writers who do this usually think the two relative pronouns are interchangeable but that which makes you look smarter. They aren't, and it doesn't. For writers, the abstract rule that that introduces restrictive elements and which introduces nonrestrictive elements is probably less helpful than the following simple test: if there needs to be a comma before the relative pronoun, you need a which; otherwise, you need that. Examples: We have a massive SUV that we purchased on credit last month; The massive SUV, which we purchased on credit last month, seats us ten feet above any other driver on the road.
The second error, even more common, is worse. It's using that when you really need who or whom. Examples: She is the girl that he's always dreamed of; Daddy promised the air rifle to the first one of us that cleaned out the hog pen. There's a basic rule: who and whom are the relative pronouns for people; that and which are the relative pronouns for everything else... The truth is that, as of 2004, misusing that for who or whom, whether in writing or speech, functions as a kind of class marker-- it's the grammatical equivalent of wearing NASCAR paraphernalia or liking pro wrestling...
Bonus Factoid and Suggestion: It so happens that you can occupy a bright child for most of a very quiet morning by challenging her to use that five times in a row in a single coherent sentence: He said that that that that that writer used should really have been a which. "