-Fully believable human performances.
-Scale
-Rich foliage, something only really seen on PC (Crysis/Far Cry)
-smoothness and immersion- fewer load times/zero pop in/higher frame rates/full HD- all whilst portraying a beautifully realised world
-Overall aesthetics, aka developers fully achieving whats in their minds and whats portrayed by concept art.

LOOOOOOOOOOOOOOOOOOOOOL

Here's one example of what is predicted to be achievable in next-gen:

Of course, I always post following when it comes to advancement of visual technology - this is rendered, and we are far away from it, but it gives a glimpse into what future holds:

But, what's probably most important in future gens is shift from polygon based worlds to voxel based. That, in combination with some sort of ray tracing (or casting) is what will bring true emulation of how our world looks and behaves to games. Of course, not all games need this, or will benefit from it, but those that are striving for realism (or quazi-realism) will be much improved once this is achieved.

There's a lot of things you could ask a more powerful piece of hardware to do.

Things like raytraced lighting for ambient occlusion , final gathering, and subsurface scattering. If they could run those things real-time, game images would look a whole lot closer to reality.

Of course personally, I'd rather see game makers use that kind of raw power for new types of stylization.

many stuff is just not as good as it should be nowadays. anti-aliasing, better textures, better lighting, draw distance, physics (dat borderlands 2 on pc) and much more.

then we have stuff like battlefield 3 which is only a dumped down pc version if you look at the multiplayer. who knows how halo would be with a better console. sure we would still have 4on4 or even 2on2 but maybe we would also have some really huge maps with 32vs32.

Speaking to GamesIndustry.biz, the executive director of production services at Ubisoft Montreal talked about challenges with the current generation of consoles. "We're extremely limited in what we can do. It's a challenge for the engineers to provide nice graphics and nice AI and nice sound with a very small amount of memory and computation time," Jacquier said. "We think that the next generation of consoles won't have these limits any more. Games might have more realistic graphics and more on-screen, but what's the value of making something more realistic and better animated if you have poor AI?"

http://www.shacknews.com/article/69172/ubisoft-procedural-ai-next-gen

Because of quotes like that from people in the industry.

Real time lighting
Texture Resolution
Interactivity
Character models
Polygon count
Reflectivity/transparency
Environmental complexity

It's not solely about visuals. I'd argue that physics and world interaction play a bigger role at this stage and will help improve immersion greatly.

The Battlefield 3 PC vs current gen consoles is actually a good comparison. The scale is one thing but it's also about having that scale whilst filling it with a fully realised world with multiple interactive objects, visual effects and complex physics. Whilst having better visuals is great, it's more in combination with the other benefits that extra power will give that we should be looking forward to.

Sticking with only visuals though, it's be great to see more titles that make proper use of tessellation and even just having higher resolution textures instead of the blurry textures we have on current gen games.

Also, it may be worth looking at some DX11 game shots to get an idea of what to expect in terms of effects:

http://www.mycrysis.com/gallery/screenshot

This gives some clue as to what some of the new visual effects will add to games but this is still only Crysis 2 so this is likely only the tip of the iceberg. Still a lot more experimentation and optimisation to be done.

Also, old but might be somewhat relevant:

http://www.eurogamer.net/articles/2011-07-20-new-xbox-and-playstation-4-what-to-expect-article