(“Trope” is from the Greek “tropos”, “manner” or “mode”. Rashi explains that the musical motifs were called trop: Talmud Kiddushin 71a, s.v. mavli’im oto.)
The “higher” chant for the Ten Commandments is more impressive and dramatic.
Some Ashkenazi communities use this version on Shavu’ot whilst on Shabbat Yitro they keep to the “lower” version.
Other congregations follow the Sephardi custom of always using the “higher” version for communal readings of the Ten Commandments and limiting the “lower” version to private study. Many Chumashim print the two versions and explain the different usage in a rubric.
There does not appear to be any detailed discussion as to the origins of the two versions, though it is generally accepted that they reflect an old tradition.
(The different customs are analysed in “Morsels of Hebrew Grammar”, the weekly Parashah sheet issued by Rabbi Meshullam Klarberg for Parashat Bamidbar/Shavu’ot 5761.)