The Jewish people mark the passage of time in two ways. We count the days of the week beginning with the six days of Creation, and we count the months starting from the Exodus from Egypt, since the Torah writes in this week's Parsha that Nissan shall be the first of all the months of the year (שמות י"ב:ב). Had the months been counted instead from Creation, as the days are counted based on creation, Tishrei would come first, because in Tishrei comes Rosh HaShanah, the Jewish New Year, that commemorates the first day of Creation.
Ibn Ezra, among other commentators, brings several proofs that Rosh HaShanah, which is observed in Tishrei, is indeed the beginning of the year counting from Creation. For example, the Torah refers to Sukkos as the harvest festival, at the end of the year (שם כ"ג:ט"ז). The harvest season is the conclusion of the outgoing year, and therefore also the beginning of the coming year; Sukkos is in Tishrei, which thus is obviously the beginning of the agricultural year. Furthermore, the beginning of the Yovel year is sanctified in Tishrei, as the Torah says (ויקרא כ"ה:ט'), and regarding the agricultural laws of that year (and the Shemittah year), the Torah says first "Do not plant" and then "Do not harvest" (שם פסוק י"א). Planting takes place near the month of Nissan, and harvesting near Tishrei. If the new year were to coincide with Nissan, then whoever would plant in the sixth year just before Nissan would be unable to harvest those crops, for the Shemittah year would be upon him.
We see from the above that Tishrei is the beginning of the year in terms of Creation and as such, in terms of laws that relate to nature and the natural order of the world. Our Parsha says, however, that Nissan shall be the first of all the months (שמות י"ב:ב'). For the Jews, then, the order of the months is determined based not on Creation, but on a different occurrence, namely Yetzias Mitzrayim. This uniquely Jewish calendar system that Hashem ordained here is hinted at by the Posuk's use of the word "לכם," for you, in presenting this Mitzvah. Hashem thus commanded the Jewish people, who left Egypt in Nissan, that they must establish a way of commemorating this event in their calendar, and thus count the months differently from the rest humanity. As Jews, we must therefore commemorate Creation through the days of the week, and the Exodus through the months of the year.
The Ramban points out that following the Babylonian exile, the months were assigned Babylonian names to remind us of the exile and of our redemption from it. Only the names of the months changed, however, while the historical events retained their significance. The Exodus and Creation thus both have an important place in the Jewish calendar established by the Torah. The Chasam Sofer says that we must therefore always use the Hebrew and not the secular calendar to mark dates in order to further our belief in the Exodus and Creation.