grammaticality - "Consider the bear that/which scratches his head." Which is correct?

If I wish to say something along the lines of

Consider the bear that scratches his head.

It seems to me that I could instead say

Consider the bear which scratches his head.

I am unsure which of these is correct, if it even matters.

Does anyone know a rule which makes this clear?

Answer

That is restrictive, it limits / restricts / specifies the identity of the subject. Using your example, the bear that scratches his head refers to one specific bear -- "the bear that scratches his head".

Which is non-restrictive, meaning it refers to something incidental about the subject. "Consider the bear, which scratches its head" refers to the bear (could be a single bear, could be the species), which happens to scratch its head.

Hope that helps!

EDIT: ShreevatsaR has pointed out that this is a convention, not a grammar rule. In the end it doesn't "matter", use the convention if it appeals to you. Here is MW's take (thanks, nohat).