At the risk of submitting a less-than-in-depth reply, I'll highlight two major inaccuracies in the hopes that someone more knowledgeable about Hamilton (though perhaps not the musical) can expand upon them -

1. The Farmer Refuted. In the musical, this is depicted as a debate in the street, with the original author reciting from a manuscript while Alexander Hamilton repeatedly interrupts. In actuality, this was a written rather than verbal disagreement, and Hamilton's responses were published anonymously as two essays - "The Farmer Refuted" and later "A Full Vindication of the Measures of Congress".

2. The years of activities leading up Hamilton writing The Reynolds Pamphlet. Here is probably where Miranda takes the most liberties with the historical record, for what I think are very good dramatic reasons. In the play, Jefferson, Madison, and Burr confront Hamilton in person about his financial improprieties. Hamilton shows them all the letters, clears his name, and then decides to author the pamphlet based on his distrust of the three men.

In actuality, Hamilton provided the letters and information regarding his affair and subsequent extortion to James Monroe and Frederick Muhlenberg, who'd been tasked with investigating his improprieties under the Monroe Commission. Monroe then sent copies of the letters to Jefferson. (Another Democratic Republican, John Beckley, is also speculated to have copied the letters.) In a 1796 essay (whose name escapes me) Hamilton made thinly-veiled references to Jefferson's own infidelities, and was rebuked a year later with references to the Reynolds allegations published in James Callender’s The History of the United States for 1796. It was a mixture of truths and falsehoods that put Hamilton in a rough spot - he had to rebuke the charges of embezzlement and treason, but couldn't rebuke all the charges of financial impropriety. So he wrote the pamphlet, which was widely disseminated by his enemies (represented in the play by Jefferson and the other Democratic Republicans showering the stage with the pamphlets, like dollar bills in a music video's strip club scene).

Ron Chernow has made some reference to this particular change in a New Yorker article published earlier this year. Miranda and Chernow maintain that the active parties were changed and the amount of confrontations and back-and-forth condensed to a single meeting to avoid introducing too many unnecessary characters and to streamline the story. The second act takes actions of various Democratic Republicans and ascribes them to different members of the party (and/or Aaron Burr, when dramatically useful) in order to keep a focused narrative.

The rest of the changes are fairly standard stuff - years of correspondence condensed, several ongoing cabinet meetings condensed into one, etc.

Slightly off-topic but worth mentioning is the way the play handles historiography and cases in which there is missing evidence or conflicting narratives, such as the deal to move the capital ("The Room Where it Happens") or Burr's insistence just before the duel that Hamilton intended to shoot to kill (because he was wearing his spectacles). The author, Lin-Manuel Miranda, is both a fan and a friend of the author Stephen Sondheim, who's written several historical musicals, among them a piece called Pacific Overtures (I've linked here to a production taped for Japanese television). The technique of having an entire song highlighting different accounts of the historical narrative, and making a point of saying "nobody knows the full account", can be traced to the Act One finale of Pacific Overtures, a number called "Someone in a Tree" that's considered in the musical theatre community to be of Stephen Sondheim's crowning songwriting achievements. Both musicals (Hamilton and Pacific Overtures) are concerned with not just the story of what happened, but about who tells that story, how we arrived at that story, what the historical record says, and how our perspectives and the perspectives of the people telling the story shape that history. In a sense, both musicals can be described as not just historical retellings, but plays about the study and reconstruction of history itself.

There's also another Sondheim musical that does this, Assassins, about various successful and unsuccessful assassins of U.S. presidents. The play is narrated by an all-American, midwestern balladeer who sings commonly-accepted descriptions of the assassins and their motivations (typically dismissing them as crazy and against the grain of American culture), while the actual depictions of the assassins show them as quintessentially American and of their time - they are not un-American, but toxic exaggerations of America.

All three musicals are excellent depictions of history as it's made, in contrast to historical musicals like 1776, which tend to depict events and characters through a narrow focus, showing clear heroes and villains and glossing over inaccuracies, gaps, and conflicting narratives.

Source: Ron Chernow's biography of Hamilton, and also Meryl Secrest's hit-or-miss biography of Stephen Sondheim.

This is a bit meta, but it may be useful to have a mega thread unpacking this and added to the FAQ, as I imagine that these questions will become pretty regular with this show being the hottest ticket on the east coast and getting so much attention.

I believe Angelica was already married when she met Hamiliton, and they had brothers, but she's such a good character.

I'm not qualified to comment on historicity, but I do know that constitutionally, the implication that Hamilton was ever able to be President in "The Reynolds Pamphlet" is incorrect.

He’s never gon’ be President now.

Never gon’ be President now.

That’s one less thing to worry about.

As the rest of the musical covers, Hamilton was an immigrant from the Caribbean, so he was never actually eligible to be president because of the Natural-born-citizen Clause.

No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President

Now some shakier ground: I've seen some things saying that that clause was written that way at the Convention in part to prevent Hamilton from ever being President. That's something I'd love to know more about from a qualified historian.