The short answer: Not really
The long answer
There are a couple main differences between $4 HDMI cables and the more expensive ones.
More expensive cables usually have more heavy-duty connectors. This is obviously important as you want to be able to reuse the cable once you buy it. Some cheap HDMI cables will break after plugging/unplugging a couple times. The actual connector, on the inside, might have bad connections to the wires and could become disconnected, therefore rendering the cable useless.
HDMI Specification Certified
The cheap cables are not usually officially HDMI Certified. The specification exists for a reason. It takes time and a lot of testing for the specification to become official. Many variables are weighed out and complexity versus quality tradeoffs are made. When a manufacturer makes a cable loosely made on a specification you might have issues with the cable. On the other hand, for a manufacturer to become certified they only have to pass the certification once. Only one cable or batch of cables might be tested. Everything else that is produced might not even go through quality control and yet still be branded certified.
Part of the certification (and HDMI spec) is a cable minimum gauge. If you are using a certified cable, everything should be ok. Companies like Monster tend to put way too much emphasis on the gauge though. Just because your HDMI cable can lift a Hummer off the ground does not mean your signal is going to be any different than a cable that uses the certified minimum gauge. A cable that is so thin that you can break it by bending it, on the other hand, will probably either stop working over time or create transmission problems.
It is commonly stated that all HDMI cables are created equal because it is a digital signal. While this is close to the truth when the spec is followed, it's not always true. The one major thing that can really destroy your signal quality is the length of the HDMI cable. It's true that HDMI signals are digital, and digital signals are 1's and 0's. The problem is that there is no such thing as a 1 or a 0 in digital electronics. It is represented in various ways. Lack/Presence of a signal, a positively magnetically charged or negatively magnetically charged medium, voltage at a certain value, etc... For instance a hard drive stores data using magnetism. The signal stored is read against an expected value range. For instance a 1 can be stored at a signal strength of 10 (while a zero is -10). A signal strength of 9.6 will also be read as 1. This is how overwritten data can be recovered. While a hard drive will read something as a definite 1 or 0. The signal strength can be used (with the help of sensitive equipment) to approximate what the previously written value was.
Here is a chart from wikipedia detailing the phenomenon:
Analog signal: +11.1 -8.9 +9.1 -11.1 +10.9 -9.1
Ideal Digital signal: +10.0 -10.0 +10.0 -10.0 +10.0 -10.0
Difference: +1.1 +1.1 -0.9 -1.1 +0.9 +0.9
Previous signal: +11 +11 -9 -11 +9 +9
How does this relate to your HDMI cable? As the length of the cable increases, not only does the signal strength decrease but so does the differentiation between each subsequent bit. If the signal quality is so bad that the machine at the other end cannot tell where one bit starts and another ends, it can guess (based on the signal strength) an incorrect value. The resulting signal is still digital, is it not? And yet it is incorrect. A poorly constructed cable’s signal degradation is affected by this problem, while high-quality cables often have active boosters. Thanks to this the signal strength stays at values that can be properly read (as a 1 and a 0) and will not blend with their neighboring bits.
I am in no way saying that you should spend $100 on an HDMI cable. I'm saying that there indeed is a difference between very cheap cables and ones that are made properly. You can definitely find HDMI cables that have reasonable prices (sometimes under $20), and are not affected by the problems I stated above. Definitely do not listen to the junk that Monster feeds it's prospective customers. Poor HDMI cables do not threaten your HDTV equipment. The cable will just not work, or work and you'll have a mosaic of boxes instead of an HD picture (like when your satellite or cable TV loses signal temporarily).
There are tons of other myths out there, like oxygen free cables. If your HDMI cable has a 1% stronger signal because of being oxygen free, how does this help you? The properly made, oxygen contaminated, HDMI spec compliant cable will still send a strong enough signal to be read as a 1 or a 0. Lets say your expected value for a 1 is 10 again. Your oxygen free cable gives you a 9.7, the other cable gives you a 9.6. Either way it's a 1.
What's the conclusion from all this? If you buy an HDMI cable, keep some things in mind. If the cable is $4 after shipping and taxes, it is probably made from the cheapest stuff the manufacturer could find. Try to buy cables that are HDMI certified or at least, knowing what you read here, guess at the quality of wires, the connectors, etc. If you are buying a long cable (over 25 feet), make sure the signal strength is kept somewhat constant between the two end points.
For those wondering, yes, I do know too much about HDMI cables.