1) The only person who can read and write English well is my friend.
2) The only person that can read and write English well is my friend.
Which one is correct? What is the difference in meaning between them?
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To answer your direct question, both sentences are correct, and there is no difference in meaning between them, so they are interchangeable.
The following English Language and Usage post has an excellent discussion, which speaks to BillJ's comment and provides further evidence in line with the comments AIQ and RubioRic provided to Sagar's answer.
The natives can tell the exact usage. But authentic grammar books say that the 2nd sentence is correct.
CGEL (page, 1251)
When the antecedent is modified by a superlative or by one of the post-determiners first, last, next, only, the relative pronoun as subject is usually that, and as object, that or zero rather than which or who(m).
Practical English Usage, Fourth Edition, 233.4
That is especially common after quantifiers like all, every(thing), some(thing), any(thing), no(thing), none, little, few, much, only, and after superlatives.
I think the link below would help you clear.
We can use 'who', 'which' or 'that'. We use 'who' for people and 'which' for things. We can use 'that' for people or things.
Personally,I think they were not different.