Living in Florida, I naturally know ALL there is to know about pulling vehicles out in snow.......
But, I have been pulling things out in nasty situations for about 40 years now, so I may have some insight.
Does that last sentence sound a little off?
Anyway, I mostly use a winch. I actually predate straps, so I am a little biased.
There are situations where only a strap will do, but in general a winch gives you more control over the situation.
The problem with using a winch is anchorage. You have to be more firmly attached than what you are trying to pull out.
I have about 3 ways to do that.
The first way, is to hook my cable up, don't start the winch.
Put the truck in 4wd, in reverse.
I can usually dig the wheels down enough to give me some hold.
In your case, unless its on pavement and you can generate enough friction to melt down to the pavement, or if the ground is not frozen deep enough for you to dig through to soft dirt, all you are going to do is polish some ice.
In that case, you are in the same situation I am in when trying to pull someone out while I am in just total owl snot.
In that case, the strap might get the job done. You can usually get enough traction to get your truck moving and let its kinetic energy pull the other one out.
The second method is to stick yourself. Many times I have backed into the ditch on the other side of the road.
Its tricky.
If the road is really bad, you need to start trying to drive yourself out just as the other car is coming up on the road.
If you time it right, both of you wind up unstuck at the same time.
If the road is good, you can wait until they are out and then use them as an anchor.
You can also use the stick-yourself trick if there is a big tree to hook to after you get them out.
The third way, is not an option for a lot of folks here with winches.
There is a reason I have a front bumper made out of 8 inch channel iron. Well, besides knocking down trees.
My winch mounts horizontally.
In other words, the front mounting point is against the inside of the front bumper.
There is nothing wrong with having the winch mount vertically on feet, but with mine, there is minimal load on the bolts.
My favorite pulling method is to put the bumper against a large tree and let the bumper do all the holding.
My transfer case is in neutral, and I do not have to hold the brakes.
In this way, there is no strain on the suspension.
That is the way I used my winch to pull the 3 ton garbage truck with all flat tires sideways across dirt at the junkyard to get parts off of it.
I was able to put my bumper against a large live oak.
The tires, and a little later the rims after the tires rolled off, dug almost down to the axle.
There is no way I could have gotten enough traction to do that pull with my tires alone.
A lot of the folks here would not have that as an option, because they kept the factory chrome bumpers.
I have deliberately left off the fourth way, but I do need to touch on it.
Using a strap or chain hooked to the rear hitch, or using a rear mounted winch to hold your truck can get nasty.
I don't know where my post is on that subject, but if you happen to be down in a dip or low spot, on a hard pull, your truck can come completely off the ground suspended between the two anchor points.
It will then most likely flip upside down like mine did and you will wind up on the ceiling with everything on the dash, seat, and floorboards in your face with the engine choking on all the oil that is suddenly in the wrong place.
If not, there is also the danger of stretching, damaging or pulling your frame in two.
My old truck, I had two heavy chains that ran from the front winch mount on both sides of the frame, to the rear trailer hitch frame.
Pulled tight.
In that way, the frame did not have to carry the load.
I realize that you were looking for strap vs winch comparisons, but you already know about straps, so I wanted to give you a winch perspective.