It seems like it would make sense to leave in time to arrive in the New World during warmer months when life is easier as you get your bearings in a new location. Did it have something to do with travel by sea? Crops? Was something going on in England that forced them out at that moment?
They were delayed by lack of volunteers to fill two ships. Then the second ship, the Speedwell, after much delay and expense proved to be unseaworthy. The logistics of getting willing "pilgrims" back from Holland and now onto one ship and problems with the British authorities plagued the entire departure.
Further delays happened when a large portion of the party decided it was too much of a disorganized mess and went back to Holland.
So poor planning, inexperience and the whole thing happening while avoiding the authorities.
A dry read, but full of information.
Edit: Specifically regarding the cold winter. They were aiming for much further south, present day Virginia. Not Cape Cod which obviously has much harsher winters.