Language complainers like William Safire and Richard Lederer often complain about the misuse of 'which' and 'that'. In school I learned the standard SAT usage of these two terms, and it made complete sense to me, because in my dialect you just didn't use these terms the way some people do. Linguists who observe the way the English language really works (as opposed to how Safire and Lederer want it to work) point out that the so-called misuse is not a misuse. It's a normal part of the English language. Arnold Zwicky even claims that virtually everyone uses 'which' in exactly the ways the style manuals say not to, including the writers of those manuals.
Well, I want to say that there is something bad about this normal part of the English language. It is a stylistic issue, and it's one that conveys something about the speaker. It's not grammatically wrong. It does sound uppity, though. In my dialect, you would never say "hand me the phone which is on the table" unless you wanted to sound like a snob. You might say "hand me the phone that's on the table". You might say, "The phone, which is on the table, is not plugged in." You wouldn't really even use 'that' unless you needed it. "Hand me the phone on the table" is much better than either, but the one with 'that' sounds ok. The one with 'which' just sounds like the kind of thing you'd expect someone with lots of money and private tutors to say.
In my generation, no one wanted to talk that way, and I went to an private school filled with rich kids. Some of their parents talked that way, and some of the teachers there might have, but I didn't really encounter this kind of thing much until graduate school, when a number of my professors did. The majority of those were from England, and the others sounded like they wanted to be. This is why I find it funny to see people like those at Language Log people talking as if there's virtually no difference between using 'which' and 'that' in these contexts. There is. It's just not a grammatical difference or difference in intended meaning (or illocutionary force, if you prefer the technical term).
The difference is like the difference between pronouncing 'ask' in the standard way or pronouncing it 'aks' as in Black English and a few other non-standard dialects. It's like the difference between pronouncing 'nuclear' the way I do or pronouncing it the way most of the country, including the president, does. These differences convey some information about the dialect someone speaks, in some cases reflecting demographics like racial groupings of geographical origin, in some cases reflecting level and kind of education of the speaker or the speaker's parents. That doesn't amount to an equivalency. It may be grammatically equivalent, but grammar isn't the only element of equivalency.
In linguists' terms, I'm not sure if you should think of this as a difference in perlocutionary force or some other pragmatic element, but it's a difference that's in the language, and I think the style manuals are giving genuinely good advice that those who don't want to sound snobbish should heed. Some may not care how they sound, but to someone of my generation in the U.S. (or at the very least in New England) who wasn't raised in a very exclusive environment, the use of relative clauses beginning with 'which' do sound a little snobbish (even if the person saying it is not a snob) in the same way that pronouncing 'nuclear' as if it were spelled 'nucular' sounds low-brow (though intellectuals in Texas will say it that way). So following this advice can have the effect of reducing how elitist someone might perceive your writing. Even if these perceptions are worth avoiding, as I would argue they are, it's a fact about how people respond to language, and we can prevent that perception by heeding some pretty simple advice.