The whole point of "registering" is that by doing so you can prove you had the material before anyone else. A publication date will accomplish this too, but in order to get something published you have to show it to a number of people first; and I tend to take Fox Mulder's line on this (Trust No One). For this reason I also would not perform un-registered songs.
If a person plagiarises another's work and the latter disputes it, they will have to be able to prove a prior claim - hence the importance of fixing the date. If each one produces a dated envelope, the earlier one wins (though of course they'll have to be opened which then destroys the seal and they can't be used after that). If the genuine writer has no proof, hard luck on them and the phony can falsely claim authorship and rip them off. That's why PROVING the date is so important. Murray's right: you automatically own the copyright on anything you create from the moment it's completed. But original work that earns royalties attracts sharks like the scent of blood, and you might have to convince the law courts that it really is yours.