For the ten days from Rosh Hashana through Yom Kipur, we add four sentences to the amida (sh'mone esre), starting respectively with "זכרנו", "מי כמוך", "וכתוב", and "(ו)בספר".
I've noticed the following practice in multiple synagogues: Between and excluding Rosh Hashana and Yom Kipur, when the leader repeats the amida, the congregation recites those sentences. However, in those same congregations, there are many people who, on Rosh Hashana and Yom Kipur, when the leader repeats the amida, recite "וכתוב" and "(ו)בספר", but not "זכרנו" or "מי כמוך".
Moreover, this practice is reflected in the ArtScroll machzor (nusach "S'farad" for Rosh Hashana and for Yom Kipur), which indicates that the congregation and leader recite "וכתוב" and "בספר" but that the leader recites "זכרנו" and "מי כמוך".
Why is the practice on Rosh Hashana and Yom Kipur different from the practice on the days between them?
I seek a reason for the difference primarily; an early source for the difference would be icing on the cake.