and what are the relative sizes and weights of the two tools?

Then I'll answer.

Backpacking - probably just bring a knife
Canoe camping gives you much more leeway weight -wise
Car camping -what ever you like

I think a small folding saw is the best combination of size weight and utility. I keep one in my whitewater kayak for emergencies. For camping and longer trips, a Sven saw is a nice if you're in a place where fires are practical and allowed.

Crow

You can't use a saw as a hammer

A saw is handy, but an axe ultimately has more usefulness.

A saw has limited functionality. The ax can chop, split, and pound.

Although I've seen far fewer injuries with the folding saw.

I'm a backpacker, and weight is at a premium. Unless the backcountry campsite has an established fire ring, I seldom build a fire. If I think I'll need to make sticks into smaller pieces that can't be easily broken and need to be sawed, I carry a small coiled wire saw that can be rigged as a bow saw with a curved green branch.

I would burn one of my solid fuel tablets if I needed a fire starter for an emergency: signaling or heat.

Like I said, I seldom make a fire when I backpack. However, if there is an established fire ring and no prohibitions, making a small fire with dead & down debris for a little warmth and enjoyment does not violate LNT ethics.

I live near Chicago, and have most often backpacked in Porcupine Mountains Wilderness State Park (upper Michigan), Red River Gorge in Kentucky, and the North Carolina Mountains. Twice in Colorado (RMNP & Lost Creek Wilderness), and twice in the High Peaks of the Adirondacks in NY state.

Care to share your backcountry haunts?

We started at the "border customs" trailhead and hiked to the ranger station campground, then to Elizabeth Lake campground where we watched an osprey soaring and fishing. On the third day, we climbed 2800 feet to Ptarmigan tunnel then descended to the Many Glacier Campground. We did the climb in clouds and a mist, so there was no sun and heat to slow us down.

The view of the red rock valley climbing Ptarmigan pass was the best. The first day of the hike was pretty super. We hiked through a high prairie of yellow flowers.

This was a trip I had wanted to do for years. I recommend it. I rode the "east side shuttle" back to the car on the fourth day.

Usually, we visit national parks and just go day hiking.

It's on the NE side of the park. Click the map rectangle-link for a special map viewer.

http://www.nps.gov/glac/planyourvisit/backcountry.htm

Backpacking, I only burn downed wood that I can break by hand or with a stomp - small enough to burn completely and not leave big partially-burned chunks - so neither would be very useful. However, I'm probably going to add a wire saw to my emergency gear this season...

If we have a gaggle going out, one axe and one Swen for the 6 of us.

My axe has a hollow handle, I keep my sharpening file there.

A saw seems a bit silly to me.