JMS's answer hits on the better reasons, this answer is mostly intended as an supplement to his.
I'm not 100% sure if the first Pilgrims truly knew how unforgiving the New World's weather would be. Much of Northern Europe and especially Britain benefit greatly from a system known as the North Atlantic Current (Or North Atlantic Drift). This pattern sees warm waters flow from the Caribbean towards northern Europe which does much to regulate its climate. Even the coldest times of winter are relatively mild all told and the temperature rarely dips below -10 Celsius (often staying above 0 for much of the year).
London's coldest month is apparently January averaging a decent 5 degrees...and it's important to note London lies around 51.5 degrees North.
North East North America actually sees the other end of this system, a cold Arctic current flowing south. New York, sitting around 40 degrees north and a significant distance south of London, sees its coldest month of January at around 2 degrees Celsius on average. St.John's Newfoundland (Canada) sits around 47.5 degrees North (still south of London) will see it's average winter temperatures drop to -8 pretty readily in January (with periods of much harsher temperatures). I would also venture a guess that the pattern now known as a Nor'Easter is a complete unknown to the Mayflower pilgrims.
So I would suggest, in addition to JMS's post, that the Pilgrims were likely unaware of how harsh a New England winter could actually be compared to their mild Northern Europe winters and as such, put less stress on arriving in the summer than they should have.