You can use an alcohol stove in the snow, and I've done so more than once --- out of necessity. You can also use the butt-end of a screw driver to pound in a nail (well, if the wood is soft enough ...). But it's not the optimal tool for that.
I think the fuel-type of a stove is at least somewhat analogous to picking a gasoline powered car vs. an electric vs. a diesel vs. hydrogen ... yes, there are variations (sometimes large variations) in various parameters between cars of the same fuel type, but the fuel type itself optimizes for certain uses.
Ditto for stoves; I think it's about knowing the major pro and con factors of each stove type to pick the best fuel source.
Alcohol stoves are pretty lightweight and can be inexpensive or essentially free, so thus I suggest a simple homemade one to new backpackers before they spend money on a stove or stove system --- that's one "pro".
The fuel source is the most widely available, especially in parts of the country that get cold enough weather so that gas stations stock gas-line antifreeze. This is why long distance hikers tend to use them. And they don't suffer from the "partial canister" problem --- you just fill your own ad-hoc plastic bottle with as much fuel as you need for a trip/stretch. And finally, you don't have an empty or (sometimes worse) partially full fuel canister to somehow dispose of at the end of your trip, a particular problem if you fly somewhere to backpack.
But alcohol is an inefficient fuel, so you need to carry more of it (or cook less); other things being equal, I find that I start out a resupply stretch (or a 'trip') somewhat heavier with an alcohol stove and at some point as I use up the fuel I break even and then end up lighter towards the end. It's also slower to cook and more fiddly to set up and get burning, you can't see the flame and sometimes can't hear for sure if it's burning or not, and as has been discussed, it doesn't work well in colder temps.
So if you backpack a lot, you might have a reason to own more than one stove type (a lot cheaper than owning several cars ... :-) ). Even if you're not a stove geek. But for most people, owning just one stove is plenty IMO, that being whatever stove best fits the type of backpacking that they do --- and --- there's a reason that canister stoves are so common.