The Chicago Manual of Style (CMOS16) ends 6.22 with this:
Although which can be substituted for that in a restrictive clause (a
common practice in British English), many writers preserve the
distinction between restrictive that (with no commas) and
nonrestrictive which (with commas). See also 5.220 under that; which.
Bryan Garner says of the distinction: “It enhances clarity. And the best American editors follow it.”
If you follow the link above and click through to the cited passage – and maybe go up a bit – you can see that Garner is clearly not happy with how “British writers have utterly bollixed the distinction between restrictive and nonrestrictive relative pronouns.”
Back to CMOS16:
In British English, writers and editors seldom observe the distinction
between the two words. See also 6.22.
That was from 5.220 under that; which.
The question is: What is the difference between the two words?
For the usage at issue, in general there is no difference. But in “polished American prose” (CMOS16/5.220) – there is.
That said, check out Which Hunting – thanks to @PeterShor for the reference – especially this:
In my research for my thesis, I’ve found that changing “which” to
“that” is the single most frequent usage change that copy editors
make. If so many writers either don’t know the rule or can’t apply it
consistently, it stands to reason that most readers don’t know it
either and thus won’t notice the difference. Some editors and
grammarians might take this as a challenge to better educate the
populace on the alleged usefulness of the rule, but I take it as
evidence that it’s just not useful.
In other words, copy editors apply the so-called rule consistently to make sure “polished American prose” is what gets published. Almost no one else notices. Or cares much if they do.