Here's how it works:
Each pixel is made up of three sub-pixels: red, green, and blue. For an LCD, since a sub-pixel is the fundamental unit, the number of bits is given in terms of per sub-pixel rather than per pixel as with a CRT. So an 8-bit LCD is the same as a 24-bit CRT (and 32-bit as well for most CRTs). This is the number of 0 or 1 values that each sub-pixel can have; essentially, then, 8-bit means each sub-pixel can display 2^8 = 256 different shades, and since there are three sub-pixels per pixel, that's 256^3 = 16777216 = 16.7M (or sometimes you'll see 16.8M) colors per pixel. For a 6-bit LCD, it's 2^6 = 64 shades per sub-pixel, and 64^3 = 262144 colors per pixel.
When the computer passes on the data to the monitor, it devotes 4 bytes per pixel. Each byte is 8 bits of data, for each of the colors. The 4th byte is actually just a dummy byte and just makes things line up (4 bytes are easier to work with than 3). For 8-bit, the values go from 0, 1, 2, ... all the way up to 253, 254, 255 (0 to 255 in increments of 1). For 6-bit, however, since they only have 64 colors to work with, the values are 0, 4, 8, etc. up to 244, 248, 252 (0 to 252 in increments of 4). To display the in-between values, then, there are two main techniques. The older method is dithering. If you wanted to display 193, the closest values are 196 and 192. Then for every 4 pixels, you set one pixel's value to 196 and the three others to 192. The eye perceives them together and it averages out to 193. The newer method is to use something called frame rate control. Essentially it's dithering but done over time instead of over space. For every 4 frames, one frame will be 196 while the other 3 frames are 192. The monitor chooses different pixels to be at 196 every frame so that you don't notice flashing of large parts of the monitor.
So with 6-bit, you lose 253, 254, 255 but can otherwise simulate the other 253 colors (0 to 252). That's 253^3 = 16194277 = 16.2M colors, which isn't as high as 16.7M, but at least it sounds a lot better than 262K. 6-bit panels are naturally faster (the reason has to do with how the pixel decays, I've seen the formula somewhere, I'll try to look it up) so there's less reliance on other techniques to speed up response time like overdrive. If colors look washed out, though, that's more likely because of incorrect color settings (i.e. color profile) I think rather than 6-bit vs 8-bit -- the main gripe with 6-bit is that they produce video artifacts (such as lines in a smooth shade gradient), not about their color gamut; aside from losing 253 to 255, they can simulate the same colors. 6-bit panels are also somewhat cheaper to make than 8-bit.
Is it important? I think it depends on the user. I thought my laptop was 8-bit for over a month because it passed the shading tests on some websites (no artifacts), until I made a 253-255 test and found it failed. It also fails the Anandtech test by the way (it looks like there's circle arcs with center of circle at the upper left), so that's a pretty good test I think (considering other shading tests failed to show that my laptop is 6-bit). Another way is to test to see if you got the values 253-255. That one can be found here:
http://www.amptron.com/chuck/bittest.bmp
If you have a TN panel, the blacks are easy to tell if you look from slightly above, while the whites are easy to tell if you look from slightly below. If you can't see the boundaries for 252-255, then 1) you have a 6-bit or 2) you're not looking at the right angle. If you can see the boundaries for both white and black, then 1) you have an 8-bit or 2) they've finally produced a 6-bit dithering scheme that can generate all 255 colors.
So for me, I used to be against 6-bit (especially with how manufacturers try to cover it up) but now I'm kinda neutral on the issue. I've been using it for about two months now and it hasn't bothered me. Then again, I don't do any graphical processing or the like, but aside from Starcraft my computer usage is mostly office and web stuff, so I've never bought into the need for 4 ms or 8 ms monitors or the like. In the end I think whether or not it's important really depends on how you use your computer. If you game (and don't really care about color quality but just need to frag that guy) then a 6-bit TN might be a good bet since it's fast and cheap. If you do color work, then you'd want an 8-bit MVA since they have better color reproduction but are generally slower. (Note: there are 8-bit TN panels as well as 6-bit MVA panels.)