why do Synthetic fibres melt and natural fibres burn?

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I was switching out the bobbin on my sewing machine, and looking for a white cotton one to match my new thread.  The problem was that I have two white threads: cotton and polyester.  Which one was in the bobbin?  I couldn't tell just by looking at it.

So I whipped out the matchbox and performed the tried-and-true burn test.  Anyone who's familiar with textiles knows that natural fibers like cotton and wool burn, but synthetic ones melt into a little blob. The thread on my bobbin burned, proving it cotton, and I went merrily on my way.

But then later I found myself asking: why do natural fibers burn but synthetic ones don't? Surely there must be a chemical reason.  Digging around on Wikipedia didn't reveal any easy answers.

First of all, let's lay down some definitions: natural fibers come from plant, animal or mineral sources. Common examples of plant fibers are cotton, linen (via flax), and hemp. Animal fibers come from a variety of mammals including sheep, goats, alpaca, and rabbits for wool and fur.  Silks are also animal fibers, and come from animals like silkworms and clams.  The most well known mineral fiber is asbestos; mineral fibers aren't used for textiles.

Man-made fibers break down into roughly two categories: regenerated and synthetic. You'll see the word artificial used as well, but it seems to have an ambiguous meaning. Anyway, synthetic fibers come from chemical reactions.  All the sources I've read talk about extrusion, but I think this is best explained with an example.

When I took organic chemistry, we synthesized nylon.  The very basic idea is that you have two chemicals, one floating on top of the other.  In between, a reaction creates nylon.  If you pull the nylon out, more nylon will be created, giving us a nylon thread. Here's a video example.  I imagine fancy machinery does this at a smaller scale to create the stuff we use regularly.

My understanding is that regenerated fiber is created with basically the same process, except one of the chemicals is a cellulose pulp from a natural source, like wood, bamboo, or seaweed.  Rayon, an "artificial silk," is one of the most common fibers of this kind, and is made from wood.

Now let's get back to the burn test.  Natural fibers burn in various ways, synthetic fibers melt, and regenerated fibers burn a little and melt a little—not terribly surprising.  But why does the distinction exist in the first place?

Cotton, linen, and other plant fibers are made mostly of cellulose, which has the ring-like structure shown below.  When cellulose burns, it takes in oxygen and puts out carbon dioxide and water.  As a balanced chemical equation: C6H10O5 + 6O2 --heat--> 6CO2 + 5H2O.  The output are gasses, so they just float away.  Ash that remains from burning anything is mostly made up of metal oxides, so I'm guessing it comes from the non-cellulose bits.
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