Though there no doubt are "puncture-resistant" tires that seek to accomplish that end by means of extra-thick rubber, for the past 15-20 years the standard for puncture resistance has been Kevlar belted tires, which look and ride like ordinary tires and weight essentially the same as ordinary tires (maybe a few grams more).
The Kevlar belt under the tread increases puncture resistance by at least a factor 10, eliminating most common glass/thorn punctures, and also reducing the tendency of the tire to "bruise".
Price wise, my recollection is that they are about 30% more expensive than standard tires, but they likely last that much longer (plus save a lot on tubes), so the cost is pretty much a wash. (The odd thing is that Kevlar-belted tires are so hard to find.)
Some people apparently confuse "Kevlar belted" tires with "Kevlar bead" tires. The latter are "folding" tires and are supposedly desired because they are a hair lighter than metal bead tires, plus they can be folded and stuffed in a pannier. But you can buy Kevlar belted with a metal bead (my preference, since folding tires are a PITA to install), and Kevlar bead tires are not necessarily Kevlar belted (in fact, most often not).
Update: Last Saturday I had my first puncture in maybe 5 years (maybe 2000 miles), while finishing up a 35-mile morning. Just got around to fixing it because it's been ungodly hot and humid. (Hitched a ride home the last two miles, since there was no way to fix the flat in 95-degree weather on a 4-foot shoulder in the sun with traffic whizzing by at 70 MPH.)
The culprit was apparently a construction staple that had pierced the tire just outside the belt, gone through the sidewall of the tube, and then punched a bigger hole in the inside diameter of the tube.
I do notice the tire is wearing a bit thin, so I'll probably have to replace it within a year or so. I'm dreading trying to find a 700/35c road tire with Kevlar belt, though -- the catalogs are now full of off-road tires and cater even less to the touring market than they did 5-10 years ago.
OK -- To make this into a direct answer to the question, the main disadvantage of Kevlar belted tires is that you forget how to change a tire, you do it so rarely.