It has everything to do with them: there's no such thing as an "adult" stem cell. There are only differentiated and undifferentiated cells, and "embryonic" is a useful description only insofar as it describes the source of origin of an established cell line. The techniques that those quotes were concerning don't destroy embryos -- they turn differentiated cells into undifferentiated cells -- but that's enough to trigger the objections.
Said another way, there's no fundamental reason that "adult" stem can't be converted to an "embryonic" state. And the quotes above illustrate that once you do that, you run afoul of the opponents. The religious groups don't want to prohibit "embryonic" stem cell research; they want to prevent anyone from doing any sort of science that they perceive to be in violation of their notion of human-being-ness. Science doesn't support the distinctions that they're making, and therefore, the conflict is unresolvable.
I am 99% sure the distinction, as it matters to the Catholic Church or anyone else using the term, is that “embryonic” refers to having been harvested from an embryo.
Said another way, there's no fundamental reason that "adult" stem can't be converted to an "embryonic" state. And the quotes above illustrate that once you do that, you run afoul of the opponents. The religious groups don't want to prohibit "embryonic" stem cell research; they want to prevent anyone from doing any sort of science that they perceive to be in violation of their notion of human-being-ness. Science doesn't support the distinctions that they're making, and therefore, the conflict is unresolvable.