In fact is by far the more common idiom. It is phrase that is used to emphasize a particular truth, especially if it is contrary to what would commonly be understood. A good synonym is actually.
In point of fact means exactly the same thing, only takes more words to do it. Most of the time it is listed in dictionaries, if it is listed at all, as in (point of) fact.
One benefit to the longer phrase is that it calls even more attention to the shared truth. However, most of the time it is simply seen as a cumbersome phrase and has been falling in its popularity since the 1800s.
A related phrase is as a matter of fact. It is still a synonym, slightly more common than the point of fact variation, but does not come close to in fact’s usage. Sometimes this is used in conversation to emphasize a differing opinion, usually in response to a wrong assumption by the other person (e.g., “As a matter of fact, I do want that cookie.”).
Note here that an alternate wording matter-of-fact, used as an adjective to mean strictly truthful and straightforward, is distinctly different than as a matter of fact.
Writing in 1908, Woodrow Wilson called it the beginning of constitutional government. But in fact, it was only one of many documents from the period, in England and elsewhere, codifying limitations on government power. [The New York Times]
In point of fact, not only do many things work better than competition, but competition doesn’t really work all that well. [Huffington Post]
“Delly’s not a dirty player at all, as a matter of fact he’s exactly the opposite,” Hunt said. [Sydney Morning Herald]