They are both means to transcribe Mandarin Chinese into Roman characters and guides on pronunciation.
Wade-Giles was developed by a Brit way back when the Brits not only conquered half the world, but studied it and named the persons, places, & things worth knowing about. Think Peking (W-G) vs Biejing (P).
Pinyin was developed in late 20th century by the Chicoms (& never adopted by the Taiwanese).
The person's goat I was getting was an example the usual, contemporary, Chinese (PRC) xenophobe/chauvinist. Of course, he thought Pinyin was the way to go, what with its being superior in every way to the "imperialist" W-G.
FWIW, I have a tin ear for tonal languages. I couldn't tell you chin, ching from ch'in by ear.