Owning both the 55-200VR and the 70-300VR and have compared not only both but two copies of the 55-200 on two Nikon DX cameras against the 70-300. Viewing was done on two hi-def monitors (25" and 27") using close to two thousand pics. Bottom line- the 70-300 blows away the 55-200. It is not even close. The MUCH heavier 70-300 is considerably sharper, colors are balanced and accurate (as they are on the 55-200) but much richer, brighter, with better contrast and much better bokeh. Skin tones and portraits in particular are wonderful with the 70-300. Downside on the 70-300 is that it is heavier. I have read some comments that the 70-300 begins losing sharpness after 200mm but my copy is super sharp at all focal lengths. If you are a serious amature and can justify the price go for the 70-300.
I owned the 70-300VR for a year and sold it. I bought the 55-200VR and still have it after a year. For me, an amateur photographer of general life pictures, I did not like the size and weight of the 70-300. I ended up not using it much and therefore decided to sell it. However, the pictures were beautiful. In it's favor are the bokeh and contrast. Otherwise, I find the little brother 55-200 more suitable for me hands down. I prefer the range, light weight, low price, and overall picture quality was more than good enough. If I were to do primarily bugs, birds type, then the 70-300 would be more suited.
Yes, the two lenses are worlds apart. The 70-300 is a top quality 'prosumer' lens, intended for advanced amateurs and pros on a budget; worth every penny! The 55-200 is an affordable, 'entry level' lens and an excellent value.
I have the 55 200 vr and the 70 300 vr. I like them both but if I had to get rid of one it would be the 55 200. There is no comparison in the build quality,, you get what you pay for. The 70 300 is heavy but I'm not even close to a pro photog and don't wag one around for long periods of time.
I have the non-VR version of both, and I personally prefer the 70-300. I got the gray market version of the 70-300 for $110...that's a deal right there. You can get that AND the 55-200mm VR and come out paying the same as you would for just the 70-300mm VR. Also, the focal range covered (55-200mm = 82.5-300mm) and (70-300=105-450mm) is much greater with the 70-300 on a DX. VR is important, yes, but with my D90 and its battery grip, I don't have much trouble getting sharp shots at 300mm and 200mm at 1/60 with either of them, respectively.
The size difference is considerable in your bag as well. The 70-300 is rather large.