Cheap HDMI cable vs. Expensive HDMI cabe, is all HDMI cables are the same

There's lots of money in cables. Your money.
Dozens of reputable and disreputable companies market HDMI cables, and many outright lie to consumers about the "advantages" of their product.
Worse, the profit potential of cables is so great, every retailer pushes high-end HDMI cables in the hopes of duping the buyer into spending tens, if not hundreds, of dollars more than necessary.
Here's the deal: expensive HDMI cables offer no difference in picture quality over cheap HDMI cables. CNET has mentioned this before, but here's the science of why.

The signal
The first thing to understand is what's transmitted over the cable in the first place. HDMI uses Transition Minimized Differential Signaling, or TMDS.
TMDS has two basic aspects. The first is that the 1s and 0s at the source (a Blu-ray player or HD cable/satellite box) are not exactly the 1s and 0s your TV uses to create a picture--at least, not in exactly the same order. Before sending the signal out via the HDMI output, the 1s and 0s are rearranged to minimize how many transitions there are. So instead of 10101010, the transmission may look like 11110000. If you really like math, how it does this is cool, but it's not really important to understanding the concept as a whole.

(Credit: HDMI.org)

Even though this conversion is weird, it makes it much more likely the data transmitted can be rebuilt on the other end (as in, at the display).
The second part of TMDS (the DS part) is the HDMI cable itself. Each HDMI cable is actually multiple, small copper wires. Two versions of the data are sent over different wires. One of these is out of phase with the "real" signal. The TV receives all the data, puts the out-of-phase signal back in phase, then compares it to the "real" signal. Any noise picked up along the way will now be out of phase, and as such it is effectively negated and ignored.
If you're an audio person, this is similar to how balanced (XLR) cables work.
TMDS works really well, allowing for short cables and fairly long cables to carry what is a pretty intense amount of data. It also means you can have inexpensive cables that work just as well as expensive ones.
More important to our discussion, it means that when something goes wrong, it goes really wrong. It's often said that with an HDMI signal, you either get everything and it's perfect, or it isn't perfect and you get nothing. In fact, I've said this. If you're getting an image that looks correct, and there are no dropouts in the audio or video, then you're getting everything that's being sent. If the cable is faulty, or it's a really long run with an under-built cable, most of the time you'll just get nothing. No picture at all.
The question I've often gotten is what if you're right on that digital precipice? That teetering space between "everything's good" and "I got nothin.'"
I'm glad you asked.
Video
As you've read, the 1s and 0s of an HD image trot happily along, more or less, from your source to your TV. Over short runs, there really isn't anything other than a faulty cable (which itself isn't that likely) that would cause any issue. Over long runs, it's possible that interference of some kind, or a poorly made cable (more on this later), can reduce the "quality" of the signal to the point where the TV can't make heads or tails of it. Heads or tails, that's a digital joke.
At this point, you're on the edge of the digital precipice. The most likely outcome is sparkles. Here's what they look like:

An HDMI cable "fail" leads to sparkles (the white pixels). Image taken on a 50-inch flat panel TV attempting to show a 1080p/60 Blu-ray concert video.

(Credit: Geoffrey Morrison)

A close-up view of the HDMI cable failure known as sparkles. Image taken from a projected image, 1080p source and display.

(Credit: Geoffrey Morrison)

Sparkles don't always have to look so extreme. Here is a less severe example. Note these are still artifacts, indicating the cable is not able to pass the intended signal. Same Blu-ray disc as above, different Blu-ray player on the same projector. The different colors in the sparkles here are due to the camera.

(Credit: Geoffrey Morrison)

It looks a lot like snow, or static. The data received by the TV wasn't enough to figure out what those failed pixels are supposed to be. Your TV likes you, though, and it really wants to show you an image. So it builds the rest of the video, minus the failed pixels.
It's important to note that this artifact is pretty unlikely, even over long runs. You are way more likely to just not get anything at all.
If it's so unlikely, why do I bring it up? Because it's important to understand that it is impossible for the pixel to be different. It's either exactly what it's supposed to be, or it fails and looks like one of the images above. In order for one HDMI cable to have "better picture quality" than another, it would imply that the final result between the source and display could somehow be different. It's not possible. It's either everything that was sent, or full of very visible errors (sparkles). The image cannot have more noise, or less resolution, worse color, or any other picture quality difference. The pixels can't change. They can either be there (perfect, yay!) or not (nothing, errors, boo!).
All the claims about differences in picture quality are remnants of the analog days, which were barely valid then and not at all valid now. There is no way for different cables to create a different color temperature, change the contrast ratio, or anything else picture quality-wise.
At this point some of you are saying "but sparkles are noise." No, I consider sparkles an example of a signal failure and as such requires a new HDMI cable. If you see sparkles, you need a different cable.
Another potential "fail" is a failure of the HDCP copy protection, which shows up as a total snowy image, a blinking image, or something else hard to miss. This is actually even less likely, as the TMDS is more likely to fail than the channel HDCP requires for its handshake. I have seen this in my testing, though, so it's worth mentioning.