When you talk about stylized art most people will think of comics/toons first, but this is just one extreme of 1000s of available possibilities to stylize art. Here are some of the most important elements:
(disclaimer: I'm no artist and this is no teaching lecture :-) )
Shape is a strong stylizing element. Most toons exaggerated this (big heads, thin legs etc.). Games like TF2 works with this element to give each character class a distinctive silhouette. Games like WoW use it in a more subtle manner, still it is there.
Color is another very strong stylizing element. As an example, toons try to use highly saturated colors, where's modern realistic games tends to use desaturated color (gray-brown). An other possiblity is the absence of color or to use a certain fixed color palett (think about movies like 300 or sin city). Take a look at Kingdoms of Armalur, an upcoming mmorpg. The general look is not really exaggerated, but the colors are highly saturated. Would you take a look at screenshots in grayscale you would consider this game more like a photorealistic approach, but the saturation of the colors shift it into the stylized space.
The absence of details helps identifying informations.The range of details reaches from small objects over surface details to color details. Toons try to get rid of almost all details, flat uni-colored surfaces with no unnecessary detail objects. Whereas photorealistic art try to get as much details as possible into its art. From a game developer view this is a most interesting element because it depends on hardware. To add detail, you need more vertices (detail objects), more surface informations (normal maps, highmaps,specular maps etcs.) and higher resolutioned surface informations(color,normal maps=>virtual texturing). In my opinion this is a major reason why games on hand-helds tend to have a stylized look, the hardware is not powerful enough.
So, why should we use more stylized art in games ? Currently there's a rush for the most realistic art in games (take a look at the latest EPIC Unreal trailer), but the really interesting part is, that the time between first show off (the wow ,amazing visuals time) to the release date (hm..visuals looks somewhat dated), often only few years, is enough to make a game look dated. Most modern games costs $XX mio. and the pressure to deliver a good looking game his very high and it seems to work for certain games, games with a low life duration (i.e. Shooter, COD ect.). On the other hand, games which costs even more over time (MMORPGS), tend to choke on the high development/maintainance costs. Take a look at Age of Conan, it looked amazing at its release time, a realistic, cool looking game, nowaday it already looks dated. The most successfully MMORPG (you know, WoW :-) ) have a heavily stylized world, but the people still love it, almost a decade later. Sure, you could argue that AoC has some major issues at the start, but WoW got similar issues.
The fact is, that people will always take a look at the visuals first. If the visuals pleased, they will be interested and keep coming back. The problem is, that photorealistic games aged too fast so that a timeout from technical issues (bugs, missing features) is often enough to keep the people away, because a better looking game has been announced or released. To get long term support from the gaming community, delivering a photorealist game is a simple top or flop story, you will never got a second chance. On the other hand a stylized game has the "chance" to unfold its potential over time. Best example is minecraft.
From an indie view we can't compete with multi-million dollar projects, so we don't have really an other option other than going for a stylized look. But on the other hand this is a major advantage because we could deliver a game and not a interactive movie. Take a look a tv-series. All the "dated" looking tv-series from the 70s,80s,90s are gone, but the simpsons are still going on.