This is not always true. When I bought plasma TV I bought the cheapest HDMI cable that I could find (a $5 cheap Asian one) to hook it up to my plasma, rationalising exactly the same as you did above: "it's a digital connection so it either works or it doesn't". Everything seemed to work well... but just *occasionally* I would have brief periods (about 20s worth once an hour) where I would see kind sparkly static all over the screen. Over the next few months I tried various combinations of things (plugged everything into a UPS, updated the firmware on my AVR, tested with a direct connection to my HTPC, tried a friend's PS3, etc) but I could not make the problem go away. Eventually all of my testing convinced me that the TV must be faulty, so I got the service people to take it away for investigation (a non-trivial task, requiring lots of phone calls and then travelling home from work on my lunch break, etc). They keep it for a week or so and can't find anything wrong with it. I, being of a cynical nature, assumed that they hadn't even turned the damn thing on and just wanted me to stop hassling them.
A few months later I'm describing this situation to my brother-in-law. He's not really as technically minded as I am, but he works in retail selling consumer electronics gear (like cameras/stereos/TVs) so he's dealt with all kinds of equipment incompatibilities and malfunctions. He tells me that he's heard of this kind of problem when using cheap HDMI cables.
"Poppycock!", I say. "Digital connections either work or they don't".
But he is insistent. He "borrows" a cable from work which he lends to me to test out his theory. Having nothing to lose (and not wanting to strain family relations) I replaced the my cheap $5 cable with the one he lent me (not a $200 monster cable but one of their cheapest which sells for around $20). To my great surprise he was right - the problem had been the cable all along! I have not seen the problem since changing the cable (this was about 18 months ago) and I have moved house twice since then.
I am aware that the plural of anecdote is not data. However, I no longer believe that all digital cables are created equal. I'm only talking about at the very very low-end of the market - the $5 cable probably didn't even meet the requirements for shielding from the HDMI specification (and the "sparkly static" was probably due to the connection to between the two pieces of equipment temporarily being degraded when it went out of range).
I don't believe for a minute that there is any difference between a $20 and $200 cable. All I am saying is that there *can* be a difference between a $5 and a $20 one. I have experienced it first hand.