SIC and NIC are both multi-layer coatings based on the same principal of using thin layers of coating. The coating thickness is 1/4 the wavelength of light, which creates destructive interference to improve transmission and reduce reflections. According to Nikon, SIC is more effective across a wider range of the spectrum, but since it is still a multi-layer coating I don't think it is a dramatic improvement. In most cases you would struggle to see the difference except that some flares are a different colour and in extreme conditions the contrast may be a little better.
In some lenses, such as the AIS 28/2.8 and 28/2, the difference between SIC and NIC coated lenses is very clear, and these are noted on my site (serial number page). On other lenses, such as the AI-S 50/1.4 the NIC and SIC coatings look very similar (maybe only a few surfaces were changed), so I was not able to determine the transition point.
SIC coatings tend to be light orange-yellow-green, while the older NIC coatings are deep purple-red/green/blue colours depending on the angle of viewing. Some NIC coated lenses have other coloured reflections such as orange and yellow, mostly from internal surfaces. NIC coating first appeared on the Nikkor-N 35/1.4 and Nikkor-N 28/2 in 1970. These early multicoated lenses tend to have deep purple-red and blue-green coatings. I imagine that Nikon continually improved their coatings, so the coloured reflections changed over time. It's also worth noting that these coatings are very thin - the thickness is just 1/4 wavelength of light - so any minor variation in thickness would change the colour of the coating. It may be that the difference is colours in the lenses above are just sample variation?