Why is Filipino spelled with an F? Philippines is spelled with a Ph. Some have said that it's because in Filipino, Philippines starts with F; but if this is so, why did we only change the beginning of Philippines?
As this article in the Spanish Wikipedia notes:
The Philippines were named for King Philip II of Spain. They were «Las Islas Filipinas», which was anglicized to the Philippine Islands.
The noun form retains the F (Filipino), while the adjective form uses Ph (Philippine Embassy).
(I've seen older texts in British English that referred to the natives as "Philippinos.")
As to why, there's this answer:
It's interesting to note that the country's official appellation in Pilipino is Repúblika ng Pilipinas. The Pilipino Express article explains that as well.
Edit: Slate has another take on Filipino/Philippines, but still doesn't explain the discrepancy in spelling between the noun and adjective forms.