The authors themselves say of the traditional terms:
They are misleading: the integrated relative is NOT always
restrictive, in the sense of picking out a subset of the set denoted
by the head noun.
They go on to give supporting examples. However, if you’re familiar with the CGEL, you probably already know this.
As Janus Bahs Jacquet states in the comments, the difference is essentially one of formality. The Cambridge Grammar of English states the following general principle:
In a wide range of informal styles, that is used instead of who/whom or
which in defining relative clauses. (p571)
This principle is confirmed by Swan in Practical English Usage:
We often use that instead of who or which, especially in an informal
style. (p478)
Swan goes on to note:
That is especially common after quantifiers like all, every(thing), something, any(thing), nothing, little, few, much, only, and after
superlatives. (p478)
When the relative reference is to a person, Swan states:
That is often used in identifying relative clauses instead of
who/whom/which. That is most common as an object or as a subject
instead of which. That can be used as a subject instead of who, but
this is quite informal. (p482)
The Cambridge Grammar of English notes (of defining/identifying relative clauses):
That may refer to the complement of a preposition, but not when the
preposition is placed immediately before the relative pronoun:
- The other girl that I told you about also lives in Bristol.
So, the following is not grammatical:
The other girl about that I told you also lives in Bristol.
It must be: ... about whom ... . Of course, this very formal usage conforms to the general principle noted above.
Best Answer
That and which are interchangeable when introducing integrated relative clauses. Although some grammar mavens (i.e., people who hold forth on such topics but know little or nothing about linguistics) and copy editors will insist otherwise, the rule is completely bogus.
See, for instance, Language Log on that vs which, written by the co-editor of The Cambridge Grammar of the English Language.
Practically speaking, it is not something that any normal person will generally notice or follow in spoken English and it's frequently — and rightly — ignored even in literary writing. So even from that point of view it's not worth worrying about.