Some say that “that” and “which” are interchangeable. But are they? Careful writers still say no, and that distinct usage of the two words is necessary to distinguish two different kinds of relative clauses.

A relative clause is a dependent clause that gives more information about the noun (or pronoun) it modifies.
Here’s an example:
The dinosaur bones, which were found last summer, were the remains of a ferocious beast.

Relative clauses fall into two types: restrictive (also called essential) and nonrestrictive (nonessential).

Restrictive clauses, as the name implies, restrict the noun. What this means is that there is a larger group that the noun could belong to, but the restrictive clause lets us know that it’s more narrowly defined.
Here’s an example:
The dinosaur bones that were found last summer will be on display at the university museum.
This implies that the specific set of bones found last summer — and not necessarily any others — are the bones that we’ll be able to see at the museum.
Restrictive clauses use “that,” and there’s no comma before the “that.”

Nonrestrictive clauses simply give us more information about the noun. That’s why they are sometimes called “nonessential” — because the information is not essential to the main point of the sentence.
Here’s an example:
The dinosaur bones, which were found last summer, will join the museum’s permanent collection.
In this sentence, the finding date is incidental to what’s going on.
Nonrestrictive clauses use “which,” and commas set off the clause — one before the “which” and one at the end of the clause.

Yes, this is pretty picky. But there are times when using the wrong clause can give a reader the wrong idea.
For example, if you say:
The museum wanted the fossils that were more than 150 million years old.
it means that the museum wanted only the ones of a certain age.
But if you say:
The museum wanted the fossils, which were more than 150 million years old.
it means the museum wanted the fossils, and they happened to be that old.

Things get even more complicated when your relative clause is introduced with a preposition. Then you use “which” no matter what.
A collector wanted the fossils on which the museum had already bid.

Punctuation becomes even more crucial when you’re talking about people, because you need to use “who” instead of both “which” and “that.”
For example, if you say:
My sister who works at the museum is coming to visit.
it implies that you have more than one sister, and the one who works at the museum is the one who’s coming to visit.
But if you say:
My sister, who works at the museum, is coming to visit.
it implies that you have only one sister, who, by the way, works at the museum, and she’s coming to visit.

One last thing: This distinction is ignored in British English, in which “that,” “which” and related comma usage have pretty much gone out the window.