2011 Subaru Legacy problems while driving downhill in snow?

I'm a fairly new driver, and just purchased a 2011 Legacy back in July. I live in New England, and since I've started driving in snow, I've noticed something about it that never happened in my old car. Whenever there's snow on the road and I'm braking while going downhill- even if I'm literally going 10 mph- the car starts shuddering and the brake pedal stops working/ gets very difficult for me to press it. Has anyone else experienced this? Hoping someone with more car knowledge can explain this to me!

Added (1). That makes sense… My old car was an 04 Honda Pilot which I assume would also have had ABS. I thought that ABS only worked once you were going over a certain speed? This will happen if I'm coasting down a hill going 10mph. So if it is the ABS, I just hold the pedal down until the car stops? It doesn't feel like it's helping the car to stop… It's more like I keep rolling while trying to push the brake pedal down harder to stop in time. But the word "juddering" explains exactly how it feels.

As Jesk said. It just sounds like the ABS doing exactly as it should.
The system works by applying and releasing the brake many times a second (the 'juddering' you feel on the pedal) thus avoiding the wheels locking and you losing control.

What was your older car? Did it have ABS and traction control?

because what you are saying sounds like it could be ABS engaging because starting to slide when you brake