This could mean that, when winter comes, the “rest” of the people—meaning everyone who doesn’t have their head up in the clouds like Marlowe’s shepherd—complain of the difficulties associated with the coming season.
It could also be a musical reference, however, to a “rest” or pause in the playing of the philomel or the singing of the nightingale, a musical silence that contrasts with the “melodious birds” we find in the second stanza of Marlowe’s poem.