Instability snow showers are typically scattered to broken in an otherwise clear sky, but if you fly into the snow column, forward visibility will be near zero. You can see straight down, but not the horizon, so VFR conditions are gone.
Snow is not particularly hazardous for airplanes in flight because it will not stick to the airframe. On most airplanes, a small strip of snow will adhere at the stagnation point on the leading edges of the wings and tails, and many pilots believe that is airframe ice. It's not. The air is dividing to flow over and under the surface at that point, so a small amount of the snow just stays there and doesn't build.
Snow can be mixed in with unfrozen supercooled water droplets that do create airframe ice, but that is not the norm. Usually the atmospheric conditions that create snow freeze all moisture available. I don't know how many times I have flown from an area of rain into snow without accumulating airframe ice. I don't know how or why that can happen. It seems like there would need to be a transition zone between rain and snow where the supercooled droplets would be present to make ice, but that's not usually what happens.
An issue when flying through snow, particularly very dry cold snow, is precipitation static. P-static, as it's called, forms when the snow particles rebound after impact with the airframe. This process leaves behind a small electrical charge, and soon the entire airplane has a different electrical potential than the air around it.
The first sign of P-static will be noise in the com radios and loss of VOR reception, or at least an unsteady VOR indication. GPS reception usually hangs on pretty well as the P-static builds, but even it can be lost in a severe encounter.
The windshield is an excellent P-static collector because it is nonconductive, so the charge can't flow toward the trailing edges and static wicks to be dissipated back into the atmosphere. If you want to experiment, hold your hand near the windshield when flying in snow. If the charge is small, you can feel the hairs on your arm stand up. If the charge is significant, your finger will get zapped. A really big charge makes Saint Elmo's fire with the spooky greenish glow, usually forming at the edge of the windshield. I have seen arcs between the windshield and the whiskey compass, particularly at night.
P-static isn't particularly dangerous in itself, but it can become a hazard if it persists, because of lost communication and navigation capability. The only solutions are to change course to exit the snow, or to try a different altitude where the snow is less dense; or the character of the snowflakes may be different enough to not form P-static.
Snow and ice are a problem in the air, but when they collect on the runway, the slippery conditions are a hazard for all pilots. Sliding off the runway on landing is an obvious threat, but on takeoff the lack of traction can send you off the runway if you have to abort, or send you sliding off the side of the pavement in a gusting crosswind when airspeed is below control effectiveness.
The worst aspect of slippery runways is that there is no uniform way to measure and report the conditions. Controllers ask pilots for braking action reports, and we respond with the terms "good," "fair" or "nil." But what one pilot experiences as fair could turn out to be nil for another.
A reason for the differences in braking action reports is often the airplane itself. A jet with a good antiskid braking system may find the stopping action to be fair, but the pilot of a piston airplane with basic brakes and no antiskid protection may have a totally different experience. The size and number of main gear tires, and even their pressure, can make a difference in traction. An effective spoiler system that kills most residual lift from the wing also makes a big difference in braking action for airplanes that have the spoilers versus those that do not.
In any case, braking reports are not an issue for most general aviation pilots because many use airports without control towers, so there is nobody to pass on the reports. When you see ice or snow on the runway at an uncontrolled and unbusy airport, there is nobody to ask and it's up to you to decide whether to attempt a landing.
When you believe the runway surface might be slippery, it is prudent to have at least twice the normal required runway length available. Try to touch down firmly in the normal touchdown zone and then immediately test the brakes for traction. If there is any doubt you will have enough runway ahead to stop, immediately go around.
The Bottom Line
The defense against winter's big weather worries always comes down to the same thing, which is to be more conservative and give yourself a lot more margin in every phase of flight. And on some days, particularly at airports with shorter runways or poor night lighting, the real safety solution is to simply wait for better weather. Spring, with its new set of weather challenges, is on the way.