From these descriptions, it sounds like the SINS system is just automated dead reckoning.
From my point of view, SINS is actually less than dead reckoning. DR can include inputs from things like Doppler Sonar, while SINS only uses inertial sensors.
A big advantage of SINS (from the military point of view) is that it does not emit any external signals (which might reveal your location to an enemy), or rely on inputs from external signals (which might be corrupted by an enemy). Biggest disadvantage (as mentioned) is that it's accuracy degrades substantially over time.
One of the systems we used to sell would allow submarines to almost surface ("periscope depth") and extend an antenna above the surface just long enough to get a satellite fix, to recalibrate their inertial system.
That was before GPS, though. Sat fixes were only available at very certain times, and it took many minutes to collect enough information to calculate a fix. Many minutes more for the calculations (with the computers available then) and after all that, results weren't good to better than 100 feet or so, for a time many minutes in the past.
Of course, the vessel would continue to move during all that, so you had to store where you thought you were, and compare that to where the satellite fix thought you were, then subtract that from where you thought you were now, to arrive at where you think you might be now.
Throw in a couple of radio nav aids and it really got fun! Loran, Omega, Syledis, etc all had errors that were worse in some directions than in others (as did the Transit sat fixes as well). So our main product was the computer system to try to arrive at the best solution given all those disparate inputs.
Good times, lots of fun problems to solve. I often worked late into the night, because it was more fun than watching television or going to the beach.