Though this isn't really a fair comparison, my Sigma 18-125mm doesn't have HSM or OS, in many other ways these lenses are quite similar. Both have roughly the same focal length and the same maximum aperture, and are aimed as a relatively inexpensive all round zoom. The Sigma however, is significantly smaller; being lighter, thinner and shorter; which has big benefits if you are intending to take it as a travel lens. Interestingly the direction of zooming is the opposite for each lens - counter clockwise on the Sigma, clockwise on the Nikon. Both lenses also have a similar minimum focal distance of about 50cm.
Optically, both suffer from lots of barrel distortion at 18mm and mild pin cushion at maximum telephoto, both traits typical for this type of lens; and easily fixed in Photoshop.
|Sigma 18-125mm @ 18mm|
|Nikon 18-105mm @ 18mm|
Sharpness at maximum telephoto is quite similar and I couldn't really tell much difference. Neither will out resolve the D90's 12MP sensor at any aperture or focal length, but I did notice more consistent colour and contrast from the Nikon.
|Sigma 18-125mm @ 125mm|
|Nikon 18-105mm @ 105mm|
The other big advantage of the Nikon is Vibration Reduction, or Optical Stabilisation. Later variants of this Sigma lens have this, and it enabled me to not worry at all about shutter speed when shooting objects for this test on the Nikon. Conversely, with the Sigma I had to reposition some objects for this test to ensure no shake.
Without doubt, the Nikon is the handier lens, mostly because it has VR. Other than that, the Nikon's only other advantage is the built in SWM lens motor which lets it Auto focus on more recent Nikon consumer bodies. The extra size though is also a bit of a pain when travelling.
However, if you have an older or mid-range Nikon body that has the screw drive for focussing, the Sigma is a good deal as these run for about half the amount of a Nikon 18-105mm on the used market. You could buy this Sigma lens used with a used D50 or D70 or body and still not spend over $250, which makes it a cheap, high quality travel camera; that you don't have to worry about as much as if you took $4000 worth of Nikon D800 and Nikkor 28-200mm lens combo. It's also a much better deal than a (small sensor) compact at the same price.
So, in conclusion, if you have the money and can deal with the larger size, buy the Nikon. If not, the Sigma will work very well for you, and you can buy a 50mm f1.8 with the money you saved.