I know both of them mean 'Happy New Year' in Chinese, but which Chinese language is each of these for?

Yes, Kung Hei Fat Choi is a "variant" of the pinyin Gong Xi Fa Cai. It's a variant in that they both indicate the pronunciation of 恭喜发财. Gong Xi Fa Cai in the pinyin for the Mandarin pronunciation and Kung Hei Fat Choi represents the way the characters are pronounced in Cantonese,