Pilipino refers to a person born in the Philippines. Filipino refers to a person not born in the Philippines, but of Philippine ethnicity.
This delineation is due to the fact that there is no phonetic equivalent of the letter 'F' in any of the Philippine Islands’ indigenous languages. Spaniards who named the Philippine Islands after King Phillip II (Las Islas Filipinas) had such a sound in their language. The closest phonetic sound inhabitants in the Philippines could use was ‘P.’
Hence, people Philippine-born refer to themselves as Pilipino (they even write it as ‘Pilipino’), while those born outside the Philippines (e.g. the United States) refer to themselves as Filipino (or Filipino-American in the above case).
My dad was born in the Philippines; he’s "Pilipino". I was born in Rampart, California; I’m Filipino-American.
Person 1: What’s your nationality?
Person 2: Full-Blooded American.
Person 1: No, where’re you from?
Person 2: West Covina, California.
Person 1: No, I mean, where’re your parents from?
Person 2: Oh – the Philippines.
Person 1: Oh so they’re Filipino.
Person 2: No, They’re "Pilipino".
Person 1: That’s what I said: Filipino.
Person 2: No, my parents are Pilipino. And I’m Filipino-American.
Person 1: …?