As F.E. mentions in his comment, the possible choice of relativiser is one of the defining differences between restrictive and non-restrictive relative clauses, along with prosody and semantic shift if the relative clause is left out.
A restrictive relative clause is a relative clause that takes a more or less generic element in a phrase and narrows it down, specifying or ‘restricting’ it. Restrictive clauses are also called integrated relative clauses because they form an integral, indispensable part of the sentence: you cannot just remove the clause from the sentence without also changing the meaning of the sentence.
A non-restrictive relative clause is one that adds additional, but non-essential, information about a subject without further narrowing down or ‘restricting’ it. Non-restrictive clauses are set off by commas (corresponding to there usually being a pause flanking them in speech), and they are also called supplementary relative clauses because they give optional, supplementary information and are not integral to the sentence: if you remove the clause from the sentence, you change the meaning. They are also called appositive relative clauses because they function much like appositions (though this is slightly confusing, as appositions themselves can also be either restrictive or non-restrictive).
Some examples (relative clause in italics):
My friend that got married last weekend had to go to hospital yesterday.
That crime novel that I am currently reading is really good.
My friend, who actually just got married last weekend, had to go to hospital yesterday.
That crime novel, which I am currently reading for the fourth time, is really good.
In the restrictive clauses, the friend and the book are being introduced for the first time, and they therefore need some further information in order to pin down which friend or book we’re talking about.
In the non-restrictive clauses, the friend and the book are already in the ‘scope’ of the conversation, and they can be mentioned without further information—but an extra tidbit is given nonetheless, just because.
Notice how I used that as the relativiser in both the restrictive clauses, but not in the non-restrictive clause? That is because non-restrictive clauses require a relative pronoun as their relativiser (that is generally not considered an actual pronoun when relative in most current analyses of English syntax—it is a non-pronominal relativiser), i.e., one of the wh-words (who, which, when, where, etc.). Restrictive clauses, on the other hand, can take both pronominal and non-pronominal relativisers (i.e., that or wh-words) with no real difference in meaning or usage.
This explains why your Einstein sentences cannot have that as their relativiser. “Albert Einstein” is already a perfectly unambiguous subject in almost any situation, so any information given about him is supplementary, rather than necessary. In the somewhat contrived situation that you were talking about two different Albert Einsteins, though, you could make a restrictive clause and use that. You’d have to add a definite article before him, though, to make a proper name generic enough to accept a restrictive clause:
The Albert Einstein that became world-famous for his general theory of relativity was a German physicist. (The one that became a mechanic in Hull wasn’t.)
The Albert Einstein that was a German-born physicist became world-famous for his general theory of relativity. (The one that was an English-born mechanic didn’t.)