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The thickness of the cable shouldn't make any difference in video quality, only advantage I see with higher quality digital cables is higher tolerance to wear and tear... and maybe more secure connectors.
Lower AWG ratings are actually thicker, so 22 AWG is the thickest of all, and will be the most expensive. I'm using a cheap 28 AWG DVI-HDMI cable to go between my graphics card and my LG L246WP, and there is no problem. Generally you'll only need the thicker cables if you want it to be more rugged (as James said), or for very long cables.
The thicker the cable the better. Now when I say this, that does not mean that the image quality will be better with a digital signal. With digital, you either get the signal or you don't. But with a lower resistance (thicker cable), your CHANCES of signal degration are less than with a thinner cable. There is a reason why the thickness of the cable generally increases as the length increases.
If you use the thinnest cable possible and get a perfect signal, you will technically not benefit from a thicker cable. However, the thicker cable will bring you more peace of mind (unless it costs that much more, lol) and should statistically increase your chances of not receiving a faulty cable, as will be able to handle a bit more abuse.
If the prices are close within reason, I recommend the thickest cable you can get (smallest gauge number). I also recommend getting a little longer than you need if there is the POSSIBILITY that you may at some point in the future re-arrange your desk setup somehow.