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.
The relative pronoun who(m) refers to animate beings, like people or animals. Personalities, in the meaning it has in your example, means the characteristic traits of a person. A person's personality is not an animate being. You can't take it to the beach, for example, or feed it donuts. The personality of a person is an inanimate object (a thing). The relative pronoun which is used to represent inanimate things.
A defining relative clause gives us extra information which helps the listener to understand which things we are talking about:
- The man who you saw yesterday is my best friend.
Here, who you saw yesterday tells us exactly which man we are talking about. A non-defining relative clause, on the other hand, gives us extra information about something or somebody we can already identify:
- Your father, whom I've known for twenty years, is the most honest man
I've ever met.
Here, the relative clause whom I've known for twenty years, is just giving the listener extra information. It is not helping the listener understand which of her fathers we are talking about!
The relative item that can be used for both things and people/animate beings. However it can only be used in defining relative clauses, and not non-defining ones:
- Your father, that I've known for twenty years, is the most honest man
I've ever met* [wrong]
- The man that you saw yesterday is my best friend. [correct]
- The sandwich that you bought looked very tasty. [correct]
That cannot occur after a preposition that has been moved to the front of the relative clause:
The options for your examples then are:
A Jekyll and Hyde is a person who has two pesonalities, one of which is bad and the other good.
and
He invents a drug which/that can separate them. When he takes the drug, he becomes an evil version of himself, who(m) he calls Mr. Hyde.
Best Answer
There is some confusion here about how relative clauses work. See McCawley's SPHE for a good description. Here are some notes:
The antecedent of the relative pronoun in a restrictive relative clause is the entire noun phrase within which the relative clause occurs (not what the relative clause modifies). (In the preceding sentence, the antecedent of "which" is the noun phrase "the entire noun phrase within which the relative clause occurs".)
A restrictive relative clause modifies an N-bar, not a noun. (So in sumelic's example in the comment, the relative clause modifies the N-bar "someone strong".)
Restrictive relative clauses can sometimes be extraposed to the end of the verb phrase, as in "A man appeared who had a sinister red beard."
A relative clause does not in general begin with a relative pronoun, but rather with a relative expression within which a relative pronoun occurs (as for instance this very relative clause, which begins with "within which").