On the Saturday night before Rosh Hashana (or the previous Saturday night), we begin saying selichot. But on that night, we wait until after midnight to say selichot, "due to the holiness of Shabbat" (Shaarei Teshuva 581:1).
The question arises: After saying havdalah, we are permitted to do all melacha. But at the same time, we are not supposed to say selichot. Melacha is apparently a much more serious violation of Shabbat than saying selichot, so why should its prohibition after Shabbat end sooner?
The answer may be that selichot do not interfere with the holiness of Shabbat. Rather, Shabbat interferes with the atmosphere of selichot.
On Shabbat we reach, and accurately feel, a higher level of spiritual inspiration than during the week. In contrast, the starting point of selichot is "kedalim ukerashim dafaknu delatecha" - that we are worthless and without spiritual accomplishments. We must feel that we are at a low point in order to motivate ourselves to strive for a higher point. Only after the special atmosphere of Shabbat has dissipated is it possible to feel this.
(Inspired by R' Amital zt"l, Alon Shevut Bogrim 4:55-59.)