Why should the voice of a woman be viewed as being different (according to Jewish Law) than that of a man, for the sake of "singing in public"?
You ask why a woman's voiceshould be treated differently to a man's voice. The referenced question “What is the source for a woman's voice as ervah?” provides you with the material for an answer. A woman's voice is considered as “ervah”.
The Wikipedia article on tzniut - modesty refers to dress, hair covering and the female singing voice and discusses its relevance to ervah.
It seems that the female singing voice has a similar categorisation as immodest dress or hair-covering. Evidently the male voice does not have the same qualities.
So the issue does not relate to the voice in the active service of G-d, but rather to not using the voice in order to serve G-d's will.