Difference between 'which' and 'that' in restrictive (defining) relative clauses

Excuse me if this topic has been brought up before though I couldn't find it. It seems that there are many similar topics related to both defining and non-defining clauses but there is still one question that bothers me.
I want to know about the difference between wh- pronouns (who, which, what...) vs. that in restrictive relative clause.

For example:
The man that I saw at the mall looked puzzled.
The man who I saw at the mall looked puzzled.
I omitted commas intentionally, as I'm willing to talk about defining clauses only.

The Cambridge Grammar of the English Language gives quite vague description on that and I cannot think about actual examples:

"Relative clauses are so called because they are related by their form to an antecedent. They contain within their structure an anaphoric element whose interpretation is determined by the antecedent. This anaphoric element may be overt or covert. In the overt case the relative clause is marked by the presence of one of the relative words who, whom, whose, which, etc., as or within the initial constituent: clauses of this type we call wh relatives. In non-wh relatives the anaphoric element is covert, a gap; this class is then subdivided into that relatives and bare relatives depending on the presence or absence of that."
Thanks in advance!

UPD: I know that it seems that in general there is no need to dig for this information but still I need this for my research.

Answer

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.