Am I just not widely read enough, and this is actually nothing to do with Yom Tov and Shabbat, but happens to be something to do with nusach/minhag/girsa among the different siddurim/machzorim I've used all my life? Or is there something in it i.e. one is more appropriate for Shabbat and and one for Yom Tov? If so, what the lesson?
I have occasionally noticed שים used on a Yom Tov (usually in a Siddur, not a machzor, so could be a copy/paste artefact - such as when you see יום טוב in brackets as if it is not always applicable on the non-musaf שלש רגלים amidahs, because they are clearly copy/pasted from the musaf one and didn't remove the brackets), but never תן in a Siddur on Shabbat.
I don't think I've ever seen שים in an ashkenazi siddur but I try to avoid them as I am sefardi.