Hello. FN makes the "true and original" Hi Power or P35. Browning Arms Co. imports them into the US with the Browning name on them. In recent years, FN also imported Mk III versions of the Hi Power into the US, but the FN name simply was not so well known as the Browning name. Other than rollmarks, locations of the serial numbers on the frames, and grips, the frames, slides, bbl, and other parts are exactly the same in either gun.
FN made some Hi Powers for Argentina but then Argentina began making their own under license from FN. These are the FM Hi Power pistols. Hungary's FEG has made and continues to make unlicensed copies of the Hi Power.
I do not remember the exact year, but I believe Browning began importing the Hi Power into the US sometime in the 1950's; the historians will know more about that than I do.
Speaking only for myself, if presented the opportunity to buy a Browning-marked or FN-marked Hi Power, I'd probably go with the latter. The same quality pistol with FN markings costs considerably less than the "real Browning Hi Power" when in fact, the primary difference is the name on the gun. The FN would have less trading value later, but for myself this is the way I'd go.
Best.