Footwear will be the main concern when there's snow, or ice on the ground. Nobody wants to slip and slide their way to a slow run and a potential injury.
If snow is the only thing on the ground, I will slip on a pair of my favorite trail shoes.
The Brooks Cascadia 5 (on the left), and the Montrail Mountain Masocist are perfect for when road and trail conditions get nasty.
Although these shoes are designed primarily for rugged trails, they come in handy when sidewalks turn into slippery, uneven, snow paths. I recently wore a pair of the Masochists for a 8 mile snow run, and they were comfortable right out of the box. The grippy soles on both these shoes will give you the traction needed for picking up the pace on snow. Also, the shoes drain water quickly which keeps the weight of the shoes feeling light.Inside the shoes, I would recommend a nice wool blend sock, like the Smartwool ones above. Most good non-cotton socks will help keep moisture away from the feet, and keep your toes warm even when wet. The better the moisture and friction control, the less likely you are to develope blisters.
If the conditions are ice, then it won't really matter what shoe you have. Yaktrax can be placed around whatever shoes you have on, and can be removed just as easily. The metal coils will dig into the ice to give you enough grip to resume running with confidence.
(standard pair of Yaktrax)
Clothing: I caved in last year and finally started wearing tights when I knew the temperature wasn't going to get above 32 degrees. Fashion aside, your muscles will thank you. 40 degrees and higher, I always wear shorts. I also go by the rule of thumb to dress like you would normally when it's 20 degree warmer. During a recent 15 degree, snowy, windy night run I wore my wind/water resistant Brooks LSD coat. It is a pretty light weight jacket, but I was more than warm for a nice two hour excersion. I also opted for a pair of wind resistant gloves. The only thing I wish I had was a balaclava to keep my face warmer.
Hydration when it is cold can be tricky. Obviously, your body still sweats, so you'll still need to drink 20-30 ounces per hour. However, cold temps can lead to frozen water bottles and camel backs. In order to prevent this you can buy a sleeve for your tube, since the tube is usually the only exposed part of the hydration system, or you can wear your camelback under your jacket. This will keep your fluids close to your body. If carrying a water bottle, flipping it upside down will help slow the freezing process since it will start from the top. Also, mixed drinks like gatorade will freeze slower than plain water because of the sugar and sodium.
This is a standard 20 ounce Ultimate Direction hand bottle which many runners now use. The hand strap makes it easy to carry for long durations, and there is a pocket for either keys, gels, etc. Plus, the soft rubber mouth piece will ensure you don't knock any teeth out. I normally use one of these for runs of an hour, or less(in hot weather I may need to carry two). I normally run races with this, since most ultramarathon aid stations are 4-5 miles apart.
For longer runs I will opt to wear the 70 ounce Nathan hydration vest. I have found it to be very light weight, comfortable, and the front chest pockets make accessing items very easy. There is also enough room to stash clothing, a cell phone, or food. I normally take this out on 15-25 mile training runs where water sources may be limited.
Last but not least, watch out for cars and snowplows. I hope this helped. Let me know if you have any questions, or if you have specific races you need advice on.