I looked it up the dictionary, and it says no difference. Probably you aren’t content with such a laconic answer. While 歌 is included in the list of 常用漢字, –I don’t know what to say in English. The basic Kanjis for everyday use– 唄 is out of it, which means you hardly see the Kanji in newspaper, magazines or articles except novels. You want to know more? Okay, there is no fixed rule which to use. Correctly speaking, you can use 歌 in all occasions, on the other hand, the usage of 唄 is quite limited, and it reminds us something special, something old like genuine Japanese folk songs. So when one wants to give some special impression, (s)he does use the kanji. You know the movie, Singin’ in the Rain. The Japanese title of it is 雨に唄えば. If it were 雨に歌えば, well, I don’t know. For me the same, but for some, maybe they get different impression.

Normally we use "歌". But for traditional Japanese songs, we can use "唄". There are some traditional songs called as "民謡", "長唄" and .... I forgot. In this case, you can use both "歌" and "唄”. As verb, there are 歌う, 唄う, 謡う　and 詠う, but normally we use only 歌う.