Are you asking your original saw/ax question from a survival kit standpoint or a camping standpoint?

If you're putting together a survival kit it depends on the area you'll be going. In my area (nothern midwest U.S.) you really don't need either. There should be plenty of hand breakable dead wood. We're not talking about building giant fires for survival.

If given a choice of one - I'd suggest a high quality folding saw - like the Silky or Laplander.

If you want to get a smaller packing store then I'd recommend the Pocket Chainsaw over wire-type saws.

If you're camping, then I'd recommend a bow saw and a long-handled ax. The Boy Scouts have been using that combination for a looong time with great success.

If you want a box saw that stows away, then consider a Sven Saw or a Trailblazer Take Down Bucksaw. The Sven is easier to unfold and use, but it tends to have some sharp edges in a pack. The Bucksaw's tube is less likely to cause damage.