I don't think your car a has limited slip differential. It probably has the lousy ASR system that puts the brake to the wheel that spins and is almost usless on hills in the snow. I have a '94 wagon, if the snow tires aren't 'first season' I don't go anywhere up hills in any kind of snowfall. Do you have the 'snowtire w/ chains' button on your dash? If so that is the equally lousy ASR defeat switch - gives me about 1500 rpm of engine when all tires are slipping. I guess it is a better system than the first generation 4Matic system though - doesn't cost as much to operate.
1994 E320 Wagon (Died @ 308,669 miles)
1995 E300 Diesel (194,000)
1999 E300 Turbodiesel (244,000 miles, dies at 255,000)
2006 Toyota Tundra SR5 AC 4X4 (115,000 miles)
2011 Audi A4 Avant (129,000 miles) hate this car!
BMW 330 xi 6 speed manual (100,034 miles)