Just as Jews are different in their way of life, they are also different in the way they count their days, months and years.
In the additional Torah chapter read this week, known as Parshat Hachodesh, we are reminded of the uniqueness of our method of measuring time.
Hachodesh hazeh lachem. "This month shall be for you" (Shmot 12:2) is the way this chapter begins. While other nations calculate their months and years based on the sun, the Creator commanded His chosen people to maintain a lunar calendar with the Month of Nissan as the first month because it was the month of the Exodus. This was intended to serve as a constant reminder of the great miracles which accompanied our liberation from Egyptian bondage.
But this is not the only element in the Hebrew calendar that serves as an important reminder that G-d created and runs the world. The names of the months we now use are of Babylonian origin and remind us of the redemption from Babylonian exile. And finally the days of the week have no names at all in the Jewish calendar. They are identified only by the number of days between them and Shabbat, enabling us to daily fulfill the Torah command to "Remember Shabbat and keep it holy".
Counting Jewishly preserves our identity as a people above time, a nation of Israel forever.
Disclaimer: This is not a comprehensive guide, rather a brief summary to raise awareness of the issues. In any real case one should ask a competent Halachic authority.