Advice on Sigma 70-200 f/2.8 II?

I shoot with a NIkon D3100. I currently have the 15-55 and 55-200. They seem to be great lenses. I'm trying to upgrade (on a budget) and I'm confused on what lens to get. For around $800 I can get the Sigma 70-200 f/2.8 II. I will be shooting mostly sports. The main thing I'm afraid of is that I will get the lens and not be able to tell $600 difference in the quality of photos than the lenses I have. I know I will be able to better in low light.

Watch this: http://www.youtube.com/..._EvjLvGpJ0
and make up your own mind.

The Sigma 70-200mm is an absolutely cracking lens for those on a budget (about half the price of the Nikon version, which retails at around £1200). The image quality is superb and that constant f2.8 aperture throughout the range will be ideal for sports work. You will very definitely be able to tell the difference!
I don't have one but a friend does (who uses a D40 & a D5000).

The 70-200mm f/2.8 is much better than the 55-200, that is for sure. As for being able to tell the 600$ difference that would depend. At f/5.6 the images will look roughly the same anyway. I've been in this hobby quite some time so small distortions, sharpness, saturation and color fringing would all be more apparent in the 55-200 for me.
Btw, sigma does sell a 50-150mm f/2.8, its priced around 500$ but I bet you could get it used for even less.
http://mycameras2006.com/sigma-apo-50-150mm-f28-ii-ex-dc-hsm

This is one of the best lenses Sigma makes. If you can't afford the Nikon 70-200mm f/2.8, it is a good choice. However, when I was in the market for this kind of lens, I bought the Nikon 80-200mm f/2.8 over the Sigma lens. But you can't use the Nikon 80-200mm lens as your camera does not have an in-camera focus motor.
There are a couple of advantages of such a lens.
First, it is good for portraiture as it will give you superior limited depth-of-field. Many portraiture photographers have gone to a high quality 70-200mm f/2.8 zoom for this reason.
The second is for sports. While the focus motor of the Sigma is a lot slower than the Nikon 70-200mm f/2.8, it is also less than half the price. So with that limitation, the lens would work well for sports (and the focus speed is really not that bad - it's just not as good as the Nikon).
Also, no lens is perfect. And with most lenses, if you use them at the maximum aperture opening, it can sometimes result in a loss of sharpness, especially in the corners. If you shot a photo with the Sigma at f/5.6 vs. Your 55-200 at f/5.6, the Sigma lens will be stopped down a bit, while the Nikon will be wide open. That fact alone will result in a sharper photo for the faster lens.
When I bought my Nikon 80-200mm, with the fast f/2.8 capability, I found I had to re-learn how to use the camera. When I use that lens, I almost always use single-point autofocus so that I can be sure I'm focusing on my intended subject. If you use dynamic or auto-area focus (the default) the lens can easily focus on the wrong thing and your photo may turn out blurry (at least for what you wanted to focus on).
You may find that when you first use the lens, you will get a lot of blurry photos. If that is the case, change to single-point autofocus. Then you will have to select the focus point when you take the photo, but you will at least know what the camera is focusing on.
So the lens has more capability, but you have to learn how to use that capability.
http://www.althephoto.com