I am new to photography. I got a chance to test both the lenses (Tamron AF 70-300mm F/4-5.6 Di LD and Nikon AF-S DX Nikkor 55-300 mm 1:4,5-5,6G ED VR).
The Nikon lens has VR and hence with full zoom, I was personally able to shoot clear hand-held pictures at a shutter speed of 1/15 in low light. Whereas, with the Tamron lens, I could not shoot even a single clear picture at S=1/15 in the same lighting conditions.
Clearly, the Nikon lens has an advantage during low light conditions but I am not sure if the picture quality (sharpness, colors) of Nikon is better than Tamron lens.
I was going through the samples pics at Amazon for both the lens and almost all the pics of Tamron seem better than Nikon to me (I mean, not all the Tamron users who are writing a review at Amazon for Tamron could be better photographers).
Nikon Sample Pics: https://www.amazon.de/dp/B003ZSHNCC/ref=psdc_5332032031_t2_B003YUBTIA
I am not sure if its worth to spend around 150€ extra for VR (if the Tamron lens actually takes better pictures).