Rubber compound is very important, but assuming similar rubber compound, there is a tread//snipe design that is more efficient for ice or for snow.
The general goal with ice is to maximise the contact surface while also removing as much water as possible from the surface of the ice (between the rubber and the ice.) For ice threads would maximise rubber contact (smaller gap between blocks) and have a lot of very thin sipes accross the surface to try and evacuate as much of that thin film of water as possible.
With snow, the goal is more to dig and grip into the snow as well as evacuate the snow to clear packed snow from the thread for the next tire rotation. So snow tire would generally have bigger blocks spaced further appart. similar to mud or sand tires.
It is hard to see when looking only at one tire, but if you look at a blizzak side by side with an x-ice, the difference are immediately visible. The X-ice has tighter grouped blocks and lot of tiny sipes, great for ice and dry pavement but it means that it has more of a tendency to float on soft snow than the blizzak.
Compound plays a big difference, so I wouldn't rely solely on the thread to decide if a tire is good on ice or not, but you can get a good hint when comparing tires. Talking about compounds, some brands put hard bits in their compound to grab the ice. Blizzak i believe (don't quote me on this) have little micro bubbles in theirs to help leave room to evacuate that thin layer of water on the ice. The old blizzard only had their special compound for a certain portion of the thread. As they wore down, they would reach a point where they lost of portion of their on ice effectiveness and became like cheaper winter tire. I don't know if that is still the case, but it wasn't a secret and I think they were mentioning on the website.