Both of these are slow mid range teles, good for daylight outdoor shots but not adequate for indoor low light action shots.
The 70-300mm can be used for both DX/APS C and full frame camera bodies, in addition to longer reach. If you don't plan on upgrading to full frame and don't need the extra reach, then you can save some money with the 55-200mm lens.
The 70-300mm is also better built, so it will be a better choice if you use this mid range tele a lot and if you like to shoot in rugged environment.
For other cheaper alternatives, you can look at 3rd party lenses from Sigma, Tamron, Tokina.
I am planning to buy either the nikon 55-200mm VR or nikon 70-300mm VR lens. I am not sure what is the major difference between the two lens except the 200mm - 300mm lens. The price of the former is almost half of the later. Please advice what is the difference between the two lenses and is there any other alternative budget lens for these.