Fortunately, there's not much evidence that any distinction is or has ever been made by careful writers of English. However, the more anally-retentive authorities say that there is a difference; so here, courtesy of Jack Lynch, it is:
If you are defining something by distinguishing it from a larger class of which it is a member, use “that”:
“I chose the lettuce that had the fewest wilted leaves.”
When the general class is not being limited or defined in some way, then “which” is appropriate:
“He made an iceberg Caesar salad, which didn’t taste quite right.”Clue: “which” is normally preceded by a comma, but “that” is not.
Now you might expect Auntie Beeb to be a bit uptight about English usage, but it seems that in the present hip BBC, you can let it all hang out.