Well, the STB algorithmically decodes the MPEG data stream (numbers) it receives, and out comes a numerical result, discretely represented by bits. A flat screen TV does scaling and all kinds of picture processing - all done numerically - before arriving to the numerical result that finally controls the display elements. HDMI is a digital connection which carries bits.
So, in the case of HDMI, we have something like this:
mathematics -> numbers -> HDMI cable -> numbers -> mathematics -> display element
...whereas in the case of a SCART, component, or any other analogue connection in the same context we have:
mathematics -> numbers -> D/A conversion -> analogue signals -> SCART cable -> A/D conversion -> numbers -> mathematics -> display element
Although you will hear all kind of voodoo around HDMI cables as well, the truth is that when the connection is OK and both ends are compatible with each other, it is a completely transparent connection, which means that both ends above get the same numbers, which in turn means a 1:1 perfect transfer. The analogue alternative below can't do that, it has the unnecessary D/A-to-A/D conversion steps, and the analogue transmission in cable will not be transparent even if the converters themselves were.
So it should be pretty much obvious that the analogue alternative will be worse - simply because it cannot be better - and as the signal gets more demanding, the more it will struggle even to get even. HD video will require more than four times the bandwidth of SD video. Depending on the implementations, the analogue transfer may or may not be visibly worse (I'm actually writing this on a 1680 x 1050 @ 60 Hz progressive computer monitor, connected via standard VGA cable, and the picture is perfect to the eye) - but the difference can be measured.
I can't think of a better analogy, but perhaps this could be compared to sending a cake to someone versus describing the recipe over telephone. In the latter case, the result may be practically just as good, but you can be sure it won't be the same.