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:
El vocablo «Filipinas» deriva del nombre del rey Felipe II de España.
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:
English never had a suitable equivalent for Filipino – a “Philippine,” “Philippian” or “Philippinian” probably just didn’t sound right, so English adopted the Spanish word Filipino, retaining the letter F and the suffix, “ino."
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.