To address the question as to why the bass instruments tend to play a less prominent part in many styles: this has to do with the fact that bass frequencies carry far less information than the higher ones. To say it very roughly, the range from 30-150 Hz (typical range of the fundamentals for the bass instruments) can carry at most 120 "units of information" each second, while the range from 150-750 Hz, which is exactly as wide a tonal range (on a keyboard, both intervals actually look the same width because the ratio is the same), can carry 600 such units each second. Both seems to be more then ever necessary, but of course only a fraction of this amount – let's say 1/24th – can actually be used for musical purposes. So we're left with 36 units of information/s in the melody range and just 6 such units in the bass range – which is really a bit poor. So in order to play interesting parts on the bass instruments, one needs to expand their frequency range. This can be done in many ways: all normal instruments produce not only fundamental frequencies but also harmonics, which get stronger when you play loudly or with stronger attack. A cello played fortissimo or staccato on the low strings will well cover the range from 100 to 2000 Hz and therefore be able to transmit all the information any player wants it to. Chris Squire's overdriven two-channel electric bass has a really strong midrange, and a funky slap bass even goes up to something like 10000 Hz.
This is all fine as long as the other instruments are willing to give that much space: in an orchestral cello solo, all the other instruments just step back to a mere acompainment role. Jon Anderson's voice is yet well above Squire's strong midrange, and in funk music most instruments and the vocals have a rather narrow bandwidth somewhere in the midrange, so the bass is allowed to explore the high-end.
But this is not always the case. In an orchestra, you normally want a rich, full harmonic sound, so you need not only space for the bass and melody but also for the 2nd violins and violas (which, by the way, are almost always much worse-audible than the basses) and for the wind instruments, so you cannot allow the bass instruments to occupy a great deal of the midrange. In many of the louder rock bands, there are two guitars and a singer fighting over the midrange domination while the drummer fills the high-end with white noise from crash and half-open hi-hat cymbals. In more tamely played rock and pop music, you do not want the bass to sound loud and agressive or hard and slappy, but this is almost impossible to avoid when giving the bass much midrange frequencies: as I said, these are harmonic overtones. But "harmonic" does not mean each of these frequencies goes very well with all of the others: the 11th overtone, for instance, is a really nasty frequency just above a tritonus. And the really high frequencies only ever occur when the strings hit the fretboard, which is why these frequencies are always short and percussive.
So the only thing you can do in such styles is to reduce the bass to an accompainment role in the background. This need not be uninteresting (many bands have the bass in the background, but it still plays very interesting stuff – often found in jazz), it can be quite loud (I understand that hip hop "music" usually has very loud bass parts), and it's definitely always very important. Even if you do not actively hear the bass at all, you will always hear it when it's missing.
The same applies to any instrument in a proper composition: even though no one ever hears the violas, an orchestra without violas will sound strangely hollow, or... hang on, no, that's just because we're playing in a church (for the funeral of twelve of our orchestra members, who were involved in a strange accident involving a 16-ton weight falling from the ceiling on a very convenient spot).
