How does the difference matter? If it's "Christianity only" then according to the original meaning of the religion clauses, non-Christian religions wouldn't have a right to freely practice their religion.
Does it even matter? In a sense yes; in a sense no. It doesn't matter in the sense that the question has been settled by judges and policy makers; even the most conservative jurists on the Supreme Court of the United States believe non-Christian religions are protected under the religion clauses. However, that "religion" originally meant "Christianity" is one of the central tenets of the "Christian America" thesis. I don't argue Christian Nationalists want to deny non-Christians the right to freely exercise their religion; though the Christian Reconstructionists (who are extremist and marginal) who borrow from Barton's work do. But I do sense these Christian America types argue non-Christians religions are lucky we Christians give them religious rights, because the original meaning of the Constitution holds we don't have to.
If you don't believe me, see mega-church pastor and Christian America promoter (David Barton speaks at his church) Robert Jeffress claim that "[n]o serious student of history doubts the framers of the First Amendment were referring to Christian denominations." Well, no, many serious students of history, for instance, the majority of the PhDs in history at prestigious universities, do not believe "religion" in the First Amendment meant "Christian sects." Though I have seen a few ultra-leftist "critical legal" theorists claim something like this in order to show just how unacceptable "originalism" is. [As the theory goes "rights" were intended to protect white, propertied, Protestant males only.]
But to the meat of the argument, the Constitution uses the term "religion" only once in the First Amendment and "religious" as in "no religious test" once in the unamended Constitution. My research concludes "religion" meant "religion in general" not "Christianity only," for all three clauses. Much groundbreaking work has already been done on the "no religious test" clause. This work demonstrates it did NOT refer to sectarian Christian tests only as Christian Nationalists argue, but rather abolishes all religious tests for public offices and permits, if the people so decide, the election of a non-Christian to public office. As Kramnick and Moore point out in The Godless Constitution the side that objected to the US Constitution on the grounds that it permitted non-Christians to be elected to office lost once the Constitution was ratified. However, the authors downplay the fact that their pious fears were largely assuaged by the fact that the "religiously correct" knew electors ultimately had the right to vote for "Christians only" if they so chose.
Now, so far, I've spoken of "religion" in the First Amendment and not differentiated between the Free Exercise Clause and the Establishment Clause. And I've done so for good reason. And that's because, even though the FEC and the EC deal with two different concepts, the term "religion" is used only once. As the clause reads, "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof;..." Note the term "religion" is used once in the EC and the term "thereof" is used in the FEC. The "thereof" in the FEC relates back to the term "religion" in the EC. It is logically impossible for something to qualify as a "religion" under the FEC, and not under the EC. This is a profound observation legal scholar Philip Hamburger makes at the 1 hour and ten minute mark in this video.
Hamburger notes some Wiccans have argued they are a "religion" for Free Exercise purposes, but not for Establishment Clause purposes. He correctly counters that such construction of the First Amendment is logically impossible because the term "religion" is used once for both of the clauses. It's like Siamese twins who are two distinct entities but share the same organ, like a heart. And in this case the term "religion" in the Establishment Clause is the "heart" that both the EC and FEC share.
I bring this up because I've seen some Christian Nationalists argue that non-Christian religions are protected under the First Amendment's Free Exercise Clause, but the Establishment Clause meant "Christian sects only." That construction of the Constitution's text is logically unsound and hence not viable. Some of them (the Christian Reconstructionists) might argue "religion" for BOTH clauses meant "Christianity" only. And that would pass the logical construction test; but the historical record shows non-Christians had rights under the term "religion" in the First Amendment.
The first rule in constitutional interpretation is we begin with a sound, logical construction of the text. Often the text is broad and "indeterminate," can support multiple meanings. But just because a text can support multiple meanings doesn't mean it can support anything. For instance the text of the Second Amendment clearly refers to some kind of "guns," but is indeterminate regarding whether government is permitted to distinguish between handguns and howitzers. But if someone argued the Second Amendment guarantees a right to play tennis, it would flunk the text test. Likewise arguing the Free Exercise Clause covers non-Christian religions, but the EC covers only "Christianity" flunks the test of logical construction of the Constitution's text.
Christian Nationalists are fond of offering a quotation by Joseph Story (a theological unitarian who thought unitarianism was "Christianity") to prove their point where Story noted:
The real object of the First Amendment was not to countenance, much less to advance Mahometanism, or Judaism, or infidelity by prostrating Christianity; but to exclude all rivalry among Christian sects and to prevent any national ecclesiastical establishment which should give to a hierarchy the exclusive patronage of the national government.
Now, Story's quotation may shed light on the underlying aim of the First Amendment (which may have had multiple underlying aims). However, it still cannot trump the TEXT of the Constitution, which uses the term "religion" not "Christianity" and where the term "religion" MUST by logical necessity cover the same faiths for both clauses.
So we come back to the blog where Christian Nationalist minister Robert Jeffress left a comment using Joseph Story's quotation to attempt to prove "religion" in the US Constitution originally meant "Christianity" only. The blog's host, Dr. Bruce Prescott, effectively countered with George Washington's 1790 letter to the Jewish congregation in Newport, Rhode Island:
The Citizens of the United States of America have a right to applaud themselves for giving to Mankind examples of an enlarged and liberal policy: a policy worthy of imitation. All possess alike liberty of conscience and immunities of citizenship. It is now no more that toleration is spoken of, as if it was by the indulgence of one class of people that another enjoyed the exercise of their inherent natural rights. For happily the Government of the United States, which gives to bigotry no sanction, to persecution no assistance, requires only that they who live under its protection, should demean themselves as good citizens.
Now this is one piece of many evidences. But that it came from the Father of America makes it quite compelling. This proves "religion" in the First Amendment meant more than just "Christianity" where the meaning of "religion" in the Free Exercise Clause, by logical necessity, defines the meaning of "religion" in the Establishment Clause and vice versa.