It would seem that original Minhag Ashkenaz was to use ta'am elyon only on Shavu'ot, but on shabbat parashat yitro and shabbat parashat va'etchanan to use ta'am tachton. This can be seen in the commentary of Hizkuni to Shemot 20:14, and is also the practice prescribed by Masat Binyamin, quoted in Magen Avraham 494:1.
Magen Avraham also notes the Sephardic practice, as described by R. Ya'akov ibn Haviv and R. Menahem de Lonzano, to always use ta'am elyon in public.
Interestingly, there seem to be many relatively old examples of Ashkenazim following the Sephardic custom. For example, R. Yoseph b. Moshe writes in Leket Yosher that his teacher R. Yisrael Isserlein would always use ta'am elyon for public reading, and would only use ta'am tachton during private study. Similary, R. Moshe Sofer in Responsa Hatam Sofer Yoreh De'ah 260 implies that ta'am tachton is only used for private study, with ta'am elyon used for all public reading.
As far as why one would differentiate between the reading on Shavu'ot and regular shabbatot, R. Yoseph Dov Soloveitchik, quoted in Reshimot Shi'urim Berachot 12a, suggests that the use of ta'am elyon is to, in a sense, recreate the experience of the Giving of the Torah at Mt Sinai, whereby we read each individual commandment as its own verse. The differing customs then reflect a disagreement as to whether this 'recreation' is only done on Shavu'ot, the anniversary of the Giving of the Torah, or instead whenever the Decalogue is read publically.
Finally, R. Soloveitchik notes that his grandfather, R. Haim Soloveitchik, only ever used ta'am tachton, even on Shavu'ot, as he did not want to break up verses in a different way to (what he believed was) the authentic tradition.