One of the biggest differences between all-wheel-drive vehicles and four-wheel-drive vehicles is in who decides when all four wheels get power. With all-wheel-drives, electronic sensors play this role; with four-wheel-drives, it's the driver who throws a lever or flips a switch to give power to all four wheels.
AWD is always "on," meaning that the electronic sensors can send power to all four wheels anytime they decide it's necessary.
4WD is engaged manually, and usually has a high range for highway speeds, and a low range for driving off-road or in heavy snow.
Both systems work well in slippery conditions. Indeed, both are superb, especially when compared to the performance of rear-wheel-drive vehicles on snow or ice. AWD may have the edge for suburban drivers who don't face snow and ice very often; 4WD may have the edge in places where winter comes early, stays long, and is fairly harsh. Both systems affect gas mileage because they add weight to the vehicles, and both mean more wear and tear on tires.
With AWD, cars are generally designed for luxury and style as much as for rugged performance, ranging from the Subaru Impreza at the low end to the Bentley Continental and Lamborghini Murecielago at the high end. With 4WD, cars tend to be more utilitarian; think Jeep and Ford Explorer.