My question is simple:
Do bindings around the neck or body of a guitar play any role in the sound that the instrument produces?
Binding, for both bodies and necks, is primarily to provide sacrificial material for minor dings and dents, which is why it's placed on edges and occasionally on the sides of fretboards.
Fretboard binding has the added bonus of preventing or at least hiding the edges of frets from sticking out and making fretting uncomfortable. Sometimes frets are poorly installed or more likely the fretboard wood shrinks slightly as it dries out which can make the frets stick out. The biggest problem with fretboard binding is that it can make refretting the guitar more difficult and more expensive.
Binding can be removed and replaced much more easily than the wood that it protects, which is why it exists. It also can look darn good when done right. As binding is not structural and makes up a very small portion of the mass of any guitar that has it, I'd be surprised if it had much of a noticeable effect on sound. Maybe some people who can hear (or claim to hear) different battery voltages in their tube screamer pedals (cough, Eric Johnson, cough) might have a sensitivity to a sound difference between bound and unbound, but I've never heard of such a distinction.
Personally I prefer bound bodies and unbound fretboards, most of the time.
On acoustic guitars the body binding is very much part of the construction so in that sense they make a difference to the sound in that the guitar would not be a guitar without them. Neck binding on all guitars and body binding on all guitars is a decorative feature. There's an ongoing war about how much difference (if any) wood makes to the sound of electric guitars, so I won't claim that binding has absolutely zero effect, but I would say it is of no great significance. To my ears Les Paul standards don't sound much different to Les Paul studios (I would likely fail a blind test). Standard have the binding and are all round more blingy whilst studios are more no-frill workhorses.
First off, Todd Wilcox is a bit "off" on his answer. His tonewood comparison between a Les Paul Standard and Studio is quite amusing, considering both have mahogany bodies with maple caps. Everyone would be hard pressed to tell the difference.
Now, YES--type of wood makes a huge difference in tone! Hell, even different fretboard woods produce noticeably different tones. Problem is, most people that try to "compare" tonewoods go about it all wrong--they use an amp. Try comparing without plugging in. Yep, now you can hear it.
To answer the question in the title 'Why some instruments have binding around the neck and/or body while some do not?', the answer comes down to money! Focussing only on electric guitars for the time being, it takes more skill and additional time to route the channels needed to bind a body or neck. In the case of a bound neck, it takes additional skill to produce binding 'nibs', which are the small binding caps that sit over the ends of the frets. Gibson, for example, only bound the Les Paul line from the 'standard' up, leaving the les Paul junior (and Melody Maker variants) with neither a bound neck or body. Fender didn't adopt binding until the '60s, with some Telecaster bodies being bound, alongside Jazzmaster necks, the very occasional Stratocaster neck here and there, the neck of the Jazz bass (though these weren't bound for the first few years of production) and a few of the CBS-era designs. Again this represented an additional manufacturing step, which was represented in the cost of the instrument passed over to the consumer.