Why is our calendar not divided into 13 four week months?

Because our system of months comes from the Greco-Roman calendar that didn't have weeks, while the 7 day week comes from Christianity's adoption of the Jewish week.

The Roman system of months had three special days in each month, the Kalends (the 1st, where we get the word "calendar"), the nones (the 5th or 7th, depending on the length of the month), and the ides (the 13th or 15th, and of course the most famous example is the ides of March). All the rest of the days were calculated from those (so the next to last day of September would be "Two days before the Kalends of October").

The Jewish calendar was based on the lunar year, not the solar year, though it's origins come from Babylon or earlier. Each month was 28 days, or a full cycle of the moon. So 7 was an easy breakdown of the month, one week starting at the new moon, one at the half moon, one at the full moon, and the last at the half moon again. (The Babylonian word for the full moon Sabattu is a cognate for Sabbath). So this calendar probably predates the idea of the creation in seven days.

There have been attempts to make a more unified calender, most notably the French Republican calendar (used during the revolution up to Napolean's takeover), which had 12 months of 30 days, each with 3 ten day weeks, and a 5 day holiday at the end of every year. They also tried to institute decimal time, so 10 hours in a day, 100 minutes in an hour, 100 seconds in a minute.