I am not a professional photographer, architect or geometry professor (one may me the nod to the reviews regarding the distortion, so forgive).
For me it is important what comes out in the end; a good photo!
The Nikon lens can definitely afford better mitgetestete even than the Sigma.
But now the last detail.
The Nikon lens is with 434 grams 80 grams lighter than the Sigma.
Personally, I find the heavier lens better already because of the lower risk of camera shake.
The 80 grams does not sound much, feel, in practice, but after much more, the Sigma works definitely heavier and more solid.
The whole feel of the Sigma is better.
But that is the only plus point, in all other disciplines that Nikon took a clear lead.
I've done a lot of comparison photos in identical situations.
The Nikon lens produces pictures with beautiful colors and better contrast, very extreme when shooting outdoors with blue / white sky visible - a very big difference.
The focus is always better when a touch Nikon.
Clearly visible when zooming into the picture.
The improved light sensitivity manifests itself, the Nikon always achieved a faster shutter speed with the smallest focal length than the Sigma.
A clear advantage of the Nikon is his noiseless image stabilizer, which acts perfectly in his role.
When Sigma is a constant "hum" heard (even when the image stabilizer).
That's when photographing unless bad but since my D3100 also has a (good) Full HD video mode, that's disturbing to hear in quiet environments on the recordings.
Also works in video mode, the autofocus of the Sigma fairly sluggish, or unsatisfactory.
Here also a clear victory for the Nikon.
Conclusion:
For me as a "mortal" Fotografierer with high demands on image quality, the Nikon AF-S DX Nikkor 18-105mm ED VR is definitely a very good choice.
One should always have in mind the price, before asking to high standards.
Strong buy recommendation!