If I had to buy one of those, I would buy the Sigma because I know the Nikon is less than mediocre. That said, I would not buy either one, I would buy the new Nikon 70-300mm AF-S VR lens because it is optically in a different league from the older Nikon 70-300mm G lens.
Yeah go with the Sigma as the Nikon G will not auto focus on the D60. Also as other have said the Nikon G is not a very good lens. However if you can get some more cash I would go for the APO version of the Sigma, I have that lens and its quite nice.
Remember that to get autofocus to work on your D60 you will need AF-S lenses from Nikon, HSM or motorized lenses from Sigma, or look for lenses that say they are motorized from Tamron and Tokina.
Oh and on Sigma lenses there equivalent of VR (vibration reduction) is OS, object stabilization. I believe for Tamron it's IS. None of these lenses have this so if hand holding make sure you turn up your shutter speed compensate when at longer lengths.