I think it really depends. A friend of mine drove his MY all winter last year when it was new (150 m round trip) and had no problems, he said. But then he got Nokian snow tires for this season, so maybe it wasn't quite as good as he implied. Where I live I'm a little bit of an outlier, even for my community. Road is plowed late (sometimes days later than the main roads). Just this week my neighbor's Audi spent the night stuck on our road. There's a foot of snow in my driveway because the plow hasn't been here for a day and a half.
So, yes. Different amounts and types of snow or ice. I have had (rare) slideoffs in several cars with all season tires, never with snow tires. You can usually adjust for conditions with all seasons, but you often have to drive much less aggressively with much less margin for error.
From what I've read, sudden regenerative braking causes the rear tires on Teslas to lock up, resulting in fishtailing in slick conditions. You can adjust by feathering the accelerator pedal, but good snow tires are going to reduce the likelihood of that happening much better.
