Back in the early '80s I used folding tyres and always carried a spare cover and a tube. I started cycling using tubulars on sprint rims so wa already used to carrying a full tyre so folders were a no brainer for me. Don't have a folding tyre asd a spare at the moment.
Does anybody else remember the Michelin bib range? I believe there is still a pair of matt (slick) bib covers on a road/track Stan Pike in my loft.
Tyres have traditionally been constructed using steel wire moulded into the rubber, this wire is known as 'the bead'. Advances in materials tech and the desire to reduce weight, particularly rotating weight, led to the introduction of aramid fibres to replace the steel. DuPont was the first to develop aramid fibres. They call it Kevlar. One of the side benefits of using aramid fibres as the bead is its ability as a material to sustain being bent without losing its integrity. Steel beads cannot be bent without sustaining permanent damage.
'Kevlar' beaded tires are therefore known as folding tires but that folding ability is merely a side benefit of the quest to reduce weight.
'Kevlar' is also used under the tread as a p*ncture proofing material. Such tires are also sometimes referred to as 'Kevlar' tires. Its important to distinguish between the two types, they are both 'Kevlar' tyres but use the material for differwent porpoises.
If you were carrying them as a spare when touring, the fact that they fold would be useful. Usually though, that they fold will only in-practice mean it costs the mail order company less to post them to you !
They are a bit lighter, as they have kevlar beads rather than steel wire, so e.g. 700x23 Vittoria Rubino Pro (folding) are ca 200g rather than 300g for Vittoria Rubino (non-folding).
A weight saving of 100g isn't a lot, but if you're saving that 100g at any point on the bike, doing it at the wheel rim will make the biggest difference.
But it would be one I'd want on my lightweight race bike and wouldn't be bothered about on my clunky commuter which weighs so much, has beefy wheels, panniers, guards, etc so it won't make that much odds.
From the manufacturers'/retailers' points of view, they're more expensive so presumably more profit, they take up less storage space and are far easier to post to you : no wonder they'd rather you bought them !
The decision to carry a folder could be affected by where you intend to be touring. In some countries tyres can be difficult to obtain, but for touring in the UK I would not carry a spare tyre..but that's just me.
I used to ride michelin pro 19's on campag omega aero rims - fantastic combo !
The reason I mention this is because the tyres where folding/kevlar beaded flopped around a bit but generally kept there shape - I even fixed a puncture in a pretty quick time once being round the corner from a chainy meet, so they do keep ther shape to a degree.