A Goretex jacket is a godsend but I’ve never rated Goretex trainers. In conditions that need Goretex water will inevitably enter over the top anyway, and now you have buckets on your feet!
That sounds right for water, but definitely not true for snow. I don't run when it's deep on the ground, but even a little can splash onto the uppers and then soak through. It's like running through shallow puddles half the time, not fording a stream. I know from experience - maybe a thousand miles of running in New England winters over the last few years - that Gore Tex and similar barrier fabrics avoid that problem quite well.
Those same shoes usually come with high-traction soles (both materials and tread pattern) which are also essential in those conditions. And of course they're warmer. Winter running in summer shoes would be a recipe for freezing, blisters, or injury around here.