Why are Saturn bands much fainter than Jupiter's?
Whilst Saturn's are not:
(except for the storm of course)
I would guess that this is a result of the slightly different composition (Saturn has more sulfur than Jupiter), as well as the distance from the sun. Increased proximity to the sun -> increased temperature -> increased surface energy -> more storms -> more mixing of the atmosphere. One simple interpretation of the bands is that they are the result of convective mixing in Jupiter's atmosphere (belts are downwelling, zones are upwelling). It would then make sense that Saturn has less mixing because it receives less energy from the sun. Beyond that, I think it's still a mystery.
I'll give this one a shot. Correction is welcome.
Upper atmosphere temperature.
It's not just elements that give a planet color, but the temperature of elements. When we examine what a planet looks like, we're basically talking about reflected sunlight from the planet's surface or atmosphere. With Earth, its atmosphere is transparent enough that its surface can be seen from space. However, it is the only planet in the Solar System where that is the case. The other planets have thick atmospheres, so all we see is basically just high altitude clouds.
Also, as I understand it, the pictures of planets are often enhanced to make any distinctions more apparent. So if you actually flew above Jupiter, its lines might not be quite so distinct as the pretty pictures you'll usually find on the web. Regardless, that doesn't change your question; its lines are still far, far more distinct than Saturn's are and Galileo was able to see its Red Spot, so I think it's safe to say its lines are pronounced, even if the pictures we get are doctored a bit.
So anyway, Jupiter is at the right distance from the Sun to undergo atmospheric transition from ice to gas. As Wikipedia puts it,
The upper ammonia clouds visible at Jupiter's surface are organized in
a dozen zonal bands parallel to the equator and are bounded by
powerful zonal atmospheric flows (winds) known as jets. The bands
alternate in color: the dark bands are called belts, while light ones
are called zones. Zones, which are colder than belts, correspond to
upwellings, while belts mark descending air. The zones' lighter
color is believed to result from ammonia ice; what gives the belts
their darker colors is not known with certainty
Ammonia ice, like all ice, is very reflective, so the colder bands with ice are lighter. Ammonia gas is transparent, but anyone who has ever flown over the ocean knows that if you have enough of a transparent thing (water), it has a distinct color. While the quote from Wikipedia above says the reason for the darker colors is uncertain, less ice in its uppermost atmosphere means less reflected light and darker color.
Saturn, Neptune and Uranus are far enough away from the Sun where they always have ice in their upper most atmospheres, so they have less variation in color. Earth's clouds are also primarily ice (not water vapor), so that is basically the answer. Jupiter is the right distance from the Sun for its upper atmosphere to transition and have distinct bands, some with ice, some without.
"But why are the bands in straight lines?"
This is due to the Coriolis effect. Viewed from above, the Coriolis effect creates bands that line up with the equator. Jupiter and Saturn both rotate quite quickly (9.5 and 10.8 hours respectively), so both have strong Coriolis effects.
Jupiter's atmosphere isn't just moving along those visible lines; it's circulating up from the warmer lower parts of the atmosphere to higher parts by convection. This is because there's a lot of heat to be transferred. Jupiter (along with Saturn, Uranus and Neptune) are all radiating more heat into space than they are receiving from the Sun, so while the heat from the Sun plays a role in not freezing Jupiter's upper atmosphere over warmer bands, it is the Coriolis effect that creates the bands.
Saturn has bands too (as you know). They are not as visible because both the warm bands and cold bands of Saturn are icy. See articles here and here and article with false color picture here, crediting Saturn's colors to ammonia ice.
So, for bands to form, all you need is relatively fast rotation. But for highly visible bands, the gas giant planet needs to be the right distance from the Sun and/or have the right amount of internal heat. It's all about the temperature.
I feel that I have to punctuate this with a "probably", because I can't swear that Saturn has ammonia ice all around its upper atmosphere, but I'm pretty sure that is the primary difference between Saturn's more uniform look (Uranus and Neptune too). It is primarily the ammonia that gives these planets their color, even though both Saturn and Jupiter are 99% hydrogen and Helium. Straight gas molecules (O2, N2, H2) and noble gases tend to have very little interaction with visible light.
Here's a fun article on gas giant planet's color. The science of what gives an atmosphere a specific color, however, is quite complex and above my pay-grade. Also, I also found this article interesting, though I can't swear to the accuracy of it. Saturn radiates a surprising amount of heat given that it's 1/3rd the mass of Jupiter.
Lastly, this one may provide a more accurate but less sexy picture of Saturn than we're used to seeing.
Final note, on Jupiter's dark spots and dark bands. Wikipedia said the reason for the darkness is unknown (as quoted above). While I think the lack of atmospheric ammonia ice is very much a part of the answer, here's an alternate explanation for Jupiter's dark red spot and (perhaps) to some extent, its dark bands. Jupiter gets, on average, about 3.4 times the solar radiation per square meter than Saturn gets. To add to that, it might have more extensive atmospheric mixing going on than Saturn and more organic compounds in its upper atmosphere, which, combined with UV rays, could well play a role in Jupiter's darker lines too.
@SirCumference After your edit in the first paragraph, its original meaning changed. Specifically, the earth is the only planet in the solar system with transparent atmosphere and other planets have thick atmospheres.
not a planetary astronomer, but is the fact that the atmosphere is colder imply that the clouds lie lower in the atmosphere than on Jupiter (and specifically the belts and zones lie below a layer of methane haze that obscures the belts and zones)?
@Bob, Jupiter doesn't have a surface so lower isn't easily defined. On rocky worlds, a colder temperature makes the atmosphere is more compact everything else being equal, but gravity is also a factor. You can get an approximation using the lapse rate formula and surface temperature because clouds form at freezing temperature as a rule. Titan for example is much colder than Earth but it's lower gravity and thick atmosphere extends much higher than Earth's.