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.

What are the disadvantages of "puncture proof" (or puncture resistant) tires?
Upon powering my netbook on, its screen turns into a garbled display (has black/white patterns, horizontal line patterns, screen freezes and/or wrong display position). This distortion happens even at BIOS startup, continuing to Windows startup. Occasionally, the issue starts around 3 minutes after the Windows desktop appears.

Pictures of the monitor (click on image to enlarge):



  

More pictures at: http://imgur.com/a/ArME1

Details:   


This happens even in safe mode (the last picture above is of the
netbook screen in safe mode).  
I  connected the netbook to an external monitor (through VGA) and the display in the external monitor shows up just fine. I have been able to use the netbook without any issues with an external monitor.    
Aside from the monitor, the netbook still works (I can shut it down with keyboard shortcuts) and the files that it has shared through LAN can be accessed fine. The torrent client's web interface on the netbook can also still be accessed on another computer.  
The issue sometimes happens even at BIOS startup.  
The variations of the distortion will sometimes change randomly.   
Occasionally, the non-distorted screen display will simply freeze.
The netbook didn't fall in the ground or get hit by an object.  
The netbook is mostly used as a torrent seeder and downloader. Its lid is opened and closed around 2 times a day only, but the netbook is powered on almost 24/7 (its lid is closed most of the time). It is mostly accessed through another computer through LAN and is not often used directly.  
The netbook was bought around 2 years ago.  


What are the possible causes of this? Any possible fixes or methods of repair I can look into? The netbook is now out of warranty.

Netbook details:  


Model: Samsung NP-N150
OS:  Windows 7 Starter  
Graphics Chip: Intel GMA3150  
Monitor: Screen Size 10.1", LED Backlit
Other specs here: http://www.samsung.com/us/computer/laptops/NP-N150-JA01US-specs

