First of all, it should be 歌を歌いたい and 歌を歌いたいです。 These both have the same meaning. The only difference is that adding "desu" makes it a little more polite or formal.

うたをうたう means sing a song and うたをうたいたい means wnat to sing a song. In very fomal way, うたをうたいます and うたをうたいたいです. But if you say without です or ます, it's not bad. Generally we use です and ます when we talk with older people, unknown people, and with whom we know but not close people.

～したい = wanna so, i wanna sing a song. = 歌を歌いたい。 i sing a song. = 歌を歌う。

They are the same, "Desu" makes it formal. BTW: 歌を歌いたい（です）。