To properly explain why a car with 2 wheel drive and a car with 4 wheel drive could differ in the ice and snow it's important to briefly consider why ice and snow caused problems in the first place. The key factor in the performance of a vehicle in these conditions is the vehicles inability to grip the road surface as much because of the barrier between the road and tyre which of course is ice or snow - this is called traction.
Tyres are designed with grooves and tread so that water, ice and snow spread into the tread to allow the tyre to contact the road directly - if less contact is made between the road and tyre then there's less grip which can lead to skidding and sliding.
This then poses the question, if what's needed is more surface area to make contact with the road how would 4 wheel drive make a difference? In short, it doesn't.
Having 4 wheel drive does not help in the ice and the snow and some may say it arguably makes things worse. There are a couple of reasons for this.
The most important aspect to consider when driving in the snow is traction - contact between the vehicle (tyre) and the road. Having 4WD does not improve the contact between the tyre and road because all 4 wheel drive system does is make the wheels turn and spin. 4WD doesn't make any difference to the tread or the surface area of the tyre.
The myth that 4 wheel drive helps on ice and snow is possibly perpetuated because in general vehicles with four wheel drives have generally larger tyres and greater surface area for the ice and snow to get into (the tread).
Having four wheel drive doesn't improve brake stop times in the snow and ice and why would it? A vehicle with two wheel drive has the same braking system a 4 wheel drive vehicle does so there is no advantage here. It could also be argued that heavier vehicles are even harder to stop, especially once they've started to slide especially on hills.
Four wheel drive vehicles do not improve the steering of the vehicle whatsoever. This myth is perpetuated with rally cars almost always having 4WD but the reason they appear to handle better in mud and so on is because the 4WD system helps the vehicle change direction better thanks to the increased power pushing it forward, plus these vehicles are specialised and will have special tyres and tuning.
There is only the one benefit of an AWD system in snow and ice and that is the ability to get moving in the first place. Four-wheel drive gives the vehicle more power to be able to go from stop to moving in difficult terrains such as mud, sleet, slush, ice and snow. These vehicles also have a better chance of getting you out of trouble. However, it remains to be said that the miscalculated self-confidence of owning a 4WD vehicle is likely to get you into more trouble in the first place when compared with a 2WD vehicle as suggested by the Car Crash Detective (see references).
The biggest problem with four-wheel drive vehicle in snow and ice is that drivers can become overconfident in their vehicle's ability to drive in such conditions which can lead to making mistakes. The Car Crash Detectives that pro-rata, four-wheel-drive vehicles are more likely to get into an accident in snow and ice than two-wheel drive vehicles. People driving 2WD vehicle do not assume their car is going to magically handle sleet and slush but 4WD owners might.
In adverse weather, it is best to not try and drive at all, however, if you absolutely must then these are some tips, much like what Popular Mechanics claim, any of these tips below are better than 4WD in the snow.
If you're looking for a 4 wheel drive vehicle then your best bet is to take a look at Audi Quattro, BMW xDrive, Mercedes 4matic and Land Rover. All of which are available as a business lease and personal lease.