Chuck,
In my experiences, I've had pros & cons with both.
I started out with a Sears walk-behind snowblower that my mother had won in a raffle back in 1970. I then used a John Deere 212 garden tractor with a 12 hp Koehler engine and a 37" snowthrower. (A snowthrower uses a high-speed auger to "throw" the snow out of a chute. A snowblower uses a low-speed auger to push the snow into a high-speed impeller, which "blows" the snow out of a chute.)
The John Deere 212/snowthrower arrangement worked very well for over 15 years under all conditions. The only problems for me were installing the snow chains on the rear tires and installing the rear wheel weights. I eventually just left the wheel weights on but the snow chains were a constant chore for me to put on every winter. The other lesser problem was getting the mower deck/snowthrower on & off every season. It was still a far cry from running a walk behind snowblower.
By the way: If you are looking at lawn or garden tractors and wondering which is which, look at the rear wheels. If the rear wheels have lug nuts, it's probably a garden tractor. If the rear wheels have a single nut holding the rear tire rims on the axles, it's probably a lawn tractor. Garden tractors are more rugged, typically have heavier frames, larger engines, electric or hydraulic lift and a more robust transmission. Garden tractors are more suited to snowblowing and tilling. Garden tractors are better equipped for extra implements than lawn tractors, although many lawn tractors can handle snowblowing tasks very well.
Any way I looked at it, the 48" mower deck, snowthrower & tiller implements were heavy, cumbersome, labor intensive and sometimes injurious to my fingers and knuckles. After I changed implements, I had to reconfigure the correct drivebelt/pulley arrangements for each implement. However, they were good for their time.
Then there's storage: Keeping the snowthrower properly stored during the summer & vice versa for the mower deck. If you have a garage or a large enough storage shed, that's usually not a problem. Same thing for a walk-behind snowblower. I recommend a level concrete surface from which to change implements.
Three years ago, I purchased a new John Deere X595 garden tractor with 4 wheel drive. Things have really come a long way. The implements on this tractor are driven by shafts, not belts, and I have hydraulic lifting; no more "armstrong" techniques hauling back on a lever to lift up the mower deck, snowthrower and tiller.
The implements are easier to attach and change, even though the weight of the implements are more than my older equipment. The big plus is no more snow chains, weights and skinned knuckles. The HST transmission on the X595 makes speed/direction changes effortless without shifting and changing gears. Power steering with the 42" snowblower hanging off the front of the tractor is a big plus. The hydraulics have a "float" feature that allows the snowblower to glide over uneven terrain without scalping the surface being cleaned if it isn't asphalt or concrete.
I've done both & I'd rather use the garden tractor with various implements rather than specialized equipment for each task. It seems that you are leaning towards a garden tractor & snowblower/thrower arrangement. Keep in mind ease of implement interchangeability, maintenance, belt/shaft drive features, manual or electric/hydraulic lift features, ruggedness/durability and service/parts availability. In other words, get something even your wife or kids would like to use.
Hope this helps & good luck.
