A quick look at Wikipedia shows that you need a license to publish:
Currently, there are many websites containing lyrics. This proposal, however, is controversial, as some sites include copyrighted texts offered without permission of the owner. United States. The Music Publishers Association (MPA), which represents the notes of the companies, launched a legal campaign against such sites in December 2005, MPA president, Lauren Keyser, said the free lyric network sites are "completely illegal" and wanted some website operators to be imprisoned in jail.
Thus, this is not a false opinion from Apple, they even protect you in some way from being held liable by the law firm (for using copyrighted material without permission).
As for other text apps on the AppStore: I assume they have a license. At least LyricFind does, since you can get a license to publish texts from them.
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