Why can't analog video be compressed like digital video (example inside)?
The analog TV signal was originally designed to be decoded with the smallest practical number of valves (tubes). Thus about half the signal space (30% of the signal amplitude and almost 30% of the time) is dedicated solely to synchronisation pulses easily detectable by analog valve circuitry, and the picture information is left with only the other half
Any enhancements over this original specification had to be implemented in a compatible way. Thus the colour signal is modulated on a high frequency carrier that does not disrupt the operation of the underlying black and white signal (though a really good B&W set will show it as a fine speckled pattern).
Later, other information (in the UK, PRESFAX, test signals - pulse and bar, one line of colour bars, CEEFAX/Teletext and closed captioning) were "compatibly" squeezed into nominally invisible "unused" lines during the field sync, but in practice you could see the moving dot pattern at the top of a badly aligned screen.
Compression could not be implemented in such a compatible manner ... how would you store a few lines of picture? Here's a box of tubes, have at it! When colour came along, a single line of low-bandwidth colour signal was stored in a delay line, for either delay-line PAL, or SECAM "Sequential Colour with Memory" decoders, but that wouldn't have been cheap enough before the mid 1960s. I think the delay line was a SAW (Surface Acoustic Wave) device.
In any case, signals as regular as your colour bar test pattern are too rare to be worth optimising. And if you saved some signal space on a simple picture, what would you do with it? A complex signal like a more typical picture needs the full bandwidth anyway.
An analog video signal is basically a waveform. It's 100% time based, and one frame takes a specific length of time to transfer, since that is how long the wave is.
The wave itself takes a certain amount of bandwidth, which is basically how much data is held in that wave. It's possible to reduce the amount of bandwidth required through various filtering techniques.
Analog video only really has the concept of "now" - the single pixel that is being displayed at that moment.
Conversely, a digital video signal is an interleaved data stream. One of the sub-streams is the picture stream. This is a frame-based stream, where each frame of the video is treated as an individual entity. It's this concept of the frames that allows video compression. Digital video has the concept of "this frame" rather than "this pixel", so it can compare neighbouring pixels in all 3 dimensions (not only the up/down left/right 2 dimensions of the frame, but also the third "time" dimension, comparing with past, and even future, frames).
An analog video signal can be fairly easily converted into a digital format through the use of a frame grabber. It can then be compressed just like any other digital format.
A good analogy would be audio. Compare an old audio cassette with an MP3. When you're playing a cassette the tape is moving past the read head at a set speed, and the read head converts the magnetism on the tape at that specific moment in time to a movement of the speaker.
Conversely, with an MP3, chunks of data (again, they're called frames) and decodes them into an audio waveform for playing through the speaker.
(note: this is a vastly simplified description, and as a result is completely wrong ;) )
You can compress analog video so it uses less bandwidth, at the cost of quality: slow scan television. Used to transmit live television from the surface of the moon, in blurry monochrome. These days we can have colour HD from the surface of Mars.
It's worth looking at how the various digital compression techniques work in detail, but they all rely on storing previous frames or bits of the current frame and computing based on the difference from the current frame. There are two reasons you can't really do this with analog:
there is no random access, fast, analog memory. The delay line mentioned by Brian Drummond is pretty much the only practical technology for analog memory, and it gives you the same signal at the same speed at a future time.
analog computation is bandwidth-limited and lossy. Gain-bandwidth product limits the extent to which you can speed it up.
Note that every frame of HD h264 decode will involve hundreds of millions of individual arithmetic operations. Encoding even more operations.