Answer: In this week's Torah portion we read about the Exodus from Egypt. Leaving Egypt was not just another historical event. The Exodus was the single most significant event in forming our identity as a Jewish nation. Whilst in Egypt we were nothing more than a group of individuals living in the same land but when G-d took us out He made us into a single unit called "The Jewish Nation". The Exodus from Egypt is so central to our religion that we are commanded to make a point of remembering it on a daily basis. We even mention it many times during our prayers, when we say the Shema and when we recite Kiddush.
Just before leaving Egypt, G-d instructed Moses to set up a Jewish calendar, stressing that the month in which the Jewish people would be leaving Egypt was always to be counted as the first month. The original system was therefore to count months in numeric order starting from the first month, the month in which we left Egypt. This insured that any time a person mentioned a month; he was actually fulfilling the mitzvah of recalling the Exodus. Furthermore, by recalling the Exodus from Egypt even when doing very ordinary things like referring to the month of the year, it brought one to realise how central going out of Egypt was to us as a nation.
For a similar reason, you'll find that we don't have names for the days of the week. The days are simply referred to as "The first day ", "The second day" etc. The only day that has a name of its own is Shabbat. This is because we are commanded to remember Shabbat on a daily basis and by referring to the days of the week in such a way, we are effectively constantly remembering and counting down towards Shabbat.
After the destruction of the First Temple we were exiled to Babylon. The Talmud tells us that when the Jews returned after a difficult period in Babylon, they began to call the months by the names that they were called in the land of Babylon. Just as the names of the months had originally been labelled in such a way in order to constantly recall the Exodus from Egypt, so too the returnees started referring to the months in the way they had been called in Babylon in order to constantly remind themselves that G-d had redeemed them from this second exile as well and had safely returned them to their land.
This does not mean that we no longer refer to the Exodus from Egypt through the months of the year. The scriptures that were written after the Jews returned from Babylon always refer to the months of the year by both titles - in order to insure that we constantly remember both exiles and show gratitude to G-d for all that He has done for us.