both will do fine in winter...just get decent snow tires, not those cheap ones.
i think they're very similar, its all just preference. personally, i'd get a corolla cause it looks nicer...not to mention the rims on the civic looks ridiculous. plus i'm not really into digital speedometers, i like the good ol' anologe.
corolla is a better choice personally but they both great cars. they both will handle great in winter just do regular maintence and get some good winter tires.