Last week I mentioned R' Leibele Eiger's explanation of there being no kiddush hachodesh for Tishrei because, unlike all other months, Tishrei was not preceded by a kedushas Shabbos -- Shabbos only came into existance after Rosh HaShana/Rosh Chodesh Tisrei. I should have given that idea more thought and explained it better. Putting aside the mysticism, what R' Leibele Eiger means (and it's also in the Shem m'Shmuel) is that what makes, for example, Kislev into Kislev is that fact that it comes after Cheshvan (or to be technical, after the last Shabbos in Cheshvan); what makes Teves into Teves is that it comes after Kislev. The flavor of each month comes from the accumulation of the past that preceded it. Stores can have a Chanukah sale in August (you know what I mean), but we can't really experience Chanukah in August or Kislev in August because the accumulated experiences of the preceding days and weeks that lead up to those dates haven't happened yet.
But what happens if there is no past experience to draw upon, or what happes if the past that is there is something a person would rather leave behind and discard rather than carry with him/her as a defining part of their future? That's the chiddush of Tishrei. There is no Shabbos mevorchim, there is no past that needs to get carried into Tishrei -- everyone has the opportunity to create a kedushas hachodesh, a kedushas ha'shana, from scratch -- all are welcome to apply, literally no prior experience necessary.
This idea lies at the heart of what teshuvah is all about. The Rambam writes (Teshuvah 2:2) that the ideal of teshuvah is to reach that point that Hashem himself testifies about the person that, "She'lo yashuv l'zeh hacheit l'olam," the sinner will not return to this aveirah ever again. Lechem Mishna asks how such a thing is possible -- a person has bechira! He/she always has the choice to repeat the same offense or to not do so.
Both the Steipler (in Karayna D'Igresa vol 2) and the Kozhiglover (Eretz Tzvi p.19) give similar answers. If even a small stain of an aveira remains behind, that aveira has the power to cause other aveiros -- aveirah gorreres aveirah. It's like weed in your garden -- if you cut off the top but don't dig down after the roots, the same weed will sprout all over again. What the Rambam means is if that if the original aveira is truly fixed, if the weed is uprooted completely, it's gone -- the same aveira, "ZEH hacheit," will never return. Sure, the garden may grow other weeds in the future -- a person still has bechira -- but these are new problems; they are not part of the aveirah gorreres aveirah cycle of the original flaw.
New beginnings with a clean slate -- that's Tisrei and Rosh haShana in a nutshell. Kesiva v'chasima tova.