As always, let's start with the basics. In this case, even the basics can be confusing.
The simple, four letter word "that" falls into a variety of categories. Most words, like "cat", have one task in life. Cat is a noun. It is a small, furry feline. However, "that" is many things:
a pronoun - That is my cat.
an adjective - That cat is my pet.
an adverb - My cat was that big.
a conjunction - Pet my cat so that it purrs.
The word "that" has been around since before the year 900.
The word "which" has also been around since at least 900, and it also falls into several categories:
a pronoun - Which do you prefer?
an adjective - I stopped petting the cat for a nanosecond, during which time the cat jumped off the couch.
A restrictive clause is one where the phrase narrows down the situation. Imagine you had a house full of cats. There are cats everywhere. Now you say:
The cat that had a grey stripe down its head came up to nuzzle me.
The purpose of the "had a grey stripe down its head" is to restrict the specific cat we're talking about.
Similarly, in a library full of books, you might talk about:
I glanced at the book that lay open on the table.
We are restricting which specific book we are talking about.
You'll note that there is no comma shown here. In a restrictive clause you generally shouldn't need to use a comma to set that clause apart. The clause is necessary to the understanding of the noun and is not separated from it.
Here's an example from The Hobbit by J.R.R. Tolkien:
[Gandalf] had made quite a dent on the beautiful door; he had also, by the way, knocked out the secret mark that he had put there the morning before.
So the "that" specifies this unique secret mark amongst all other marks. There is no comma used.
Another example from the same book:
The dark came into the room from the little window that opened in the side of The Hill; the firelight flickered - it was April - and still they played on, while the shadow of Gandalf's beard wagged against the wall.
Again we have a little window "that" opened. Bilbo's house has many windows. Tolkien is specifying which particular window he's talking about. There are no commas around the "that" phrasing.
A nonrestrictive clause is one where the description simply adds more details to the situation. Imagine you had a house with just one cat in it. This cat is the only cat in the entire building. Then you say:
The cat, which had a grey stripe down its head, came up to nuzzle me.
We are simply describing the cat for the audience. It's the only cat in the room, so we're not trying to distinguish the cat from any other cat.
Similarly, imagine an empty room. Imagine the room has only has one table in it and that table has one book on it. There are no books anywhere else in the house. We then read:
I glanced at the book, which lay open on the table, then turned away.
So the nonrestrictive clause is simply additional information about the object. It's not required to know the specific item we're discussing. Also, it is set apart by commas. This helps to further indicate that the information is incidental and not necessary in an understanding sense.
Here's an example from The Hobbit by J.R.R. Tolkien:
The chance never arrived, until Bilbo Baggins was grown up, being about fifty years old or so, and living in the beautiful hobbit-hole built by his father, which I have just described for you, until he had in fact apparently settled down immovably.
The "which" clause is set off by commas. It is purely incidental. We know exactly what hobbit hole Tolkien is describing here. The "which" simply adds some additional details about it.
Another example by Tolkien:
"What's he got in his handses?" said Gollum, looking at the sword, which he did not quite like.
There is only one sword there - the one in Bilbo's hand. We know exactly what sword is involved. The "which" clause here adds additional information about the sword. That information is set off by a comma.
Does That vs Which Matter?
While using that vs which used to be an enormously important deal, in modern times it is colloquially becoming more of an oxford comma vs no oxford comma issue - i.e. one that people tend to be lenient about. Maybe that's because many people aren't sure which one to use :). Or maybe it's because in many cases to distinguish between a phrase which is absolutely necessary and one which is incidental to the story can be a judgement call. Again, looking at Tolkien, he says in one spot:
Bifur and Bofur went out too, and came back with clarinets that they had left among the walking-sticks.
Why is "that" being used there? Is the clarinet phrase really a restrictive one? Is Bilbo's house stocked full of clarinets and we had to know that we were discussing the particular clarinets which were among the walking-sticks? Surely the "they had left among the walking-sticks" part is added description about where the clarinets were located.
So you can see that the "that" vs "which" is not necessarily an exact science. You do the best you can, but in the end I do believe there is some wiggle room. For me, part of how you make that determination is if you would use a comma in the location. If Tolkien had written:
Bifur and Bofur went out too, and came back with clarinets, which they had left among the walking-sticks.
That feels clunkier to me. So I prefer the first phrasing with the "that" and without the comma.
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