IIRC, the AWD systems on normal (non-STi) Subies are a viscous coupling for the manuals and a electro-mechanical clutch. They've probably got open diffs in the front and rear and use the brakes to act like a limited slip diff.
Mazda claim the CX-3's i-Activ AWD system is better because it's predictive. That said, it's still an electro-mechanical clutch and uses brakes to manage torque between wheels of the same axle. So it's really just computer programming that they say makes the difference.
AWD and snow tires serve slightly different purposes.
AWD helps you get moving in low traction conditions, especially in deep snow. It won't help you turn or stop any better.
Snow tires help you get moving in low traction as well as turning and stopping
I can verify that AWD + snow tires make the CX-3 almost unstoppable in snow as deep at 6" (probably deeper, but we haven't had snow deeper than 6" since I got my snow tires).
Whether or not you should get snow tires depends on how risk-averse you are. Driving to work on 5"-6" of unplowed interstate highway in Chicago, I was relaxed and weaving in between stuck cars at 5 mph. Others seemed a lot more stressed out.
Last year, Discount Tire Direct offered a stupidly wicked deal around President's Day and I got $320 off a set of Continental WinterContact SI tires mounted to 16" alloys. The whole packaged cost me a bit over $300 delivered. Right now, they're offering $75 instant rebate on a set of winter tires (excluding Bridgestone Blizzaks), with an extra $50 MIR on the same set if you apply for their credit card and use it. There's also $50 instant savings and a $50 MIR if you buy a set of wheels with your DTD credit card. The interest rate is ridiculous but if you just pay it off right away, you're golden.
So, right now, with the right wheel
combo, you'd be looking at (roughly) $550. Is that worth it to you?