In modern Japanese, 歌 is used to refer to almost any type of song (from pop songs to lullabies), whereas 唄 is used primarily for traditional Japanese songs (e.g., played on the shamisen (a Japanese three-stringed instrument)). If you called, say, a Lady Gaga song a 唄, you Japanese teacher likely corrected it to 歌. I hope this answers your question! :)

In Chinese 唄 means Chant and this is for song 歌. 歌 is mostly used.