Here's what I see as the difference:
"Only three domestic animals are not mentioned in the Bible" give me concrete, clear information. (It may be inaccurate, but I know what it means.)
"One of the only domestic animals not mentioned in the Bible" implies that there ARE other domestic animals not mentioned in the Bible. (As Bingley has indicated, this is indeed true.) Why not say "One of only seven domestic animals not.." or "one of only a few domestic animals not..." or even "the only one of our Western European-centric idea of domestic animals not.....", any one of which would be clearer. The "only" is unnecessary and tends towards obfuscation and bewilderment, which I eschew.

Of course, this is my own opinion, which everyone else is free to ignore.