In broad terms, the difference between INS and IRS is about 30 years of technological advance.
INS emerged in military equipments in the early sixties and in civil systems in the late sixties/early seventies. IRS emerged in the nineties as part of a bigger system - the FMS.
There are 3 main differences between INS and IRS:
1) A far greater degree of system integration. In the sixties, you probably already had a B707 or two in your fleet. It was fitted with VOR and DME. Maybe it even carried a flight navigator and a Doppler system. You bought a bolt-on INS and it did the navigation for you. Sure, it had all these wonderful gyros in it, but you still carried on using your gyro-magnetic compass and your ADI because it would have cost too much to re-design the system in retrospect.
IRS is different. The systems engineer is brought in at the design stage of the aircraft, along with the airframe designer and the engine designer. Now you use IRS gyros as your primary reference source for EFIS, for stabilising the AWR scanner, you use IRS velocities in your maxaret braking system and your FADEC, you have a vertical channel for your VSI to replace those old-fashioned dashpots, and attitude, accelerations, velocities and positions are all fed to the FDR. It's now all part of one big system.
2) The advent of ring laser gyros. Thirty years ago you had lumps of rotating metal in your INS. Now we shoot photons through a prism and measure the path difference. There are no moving parts. With INS the mean time between failures (MTBF) was of the order of 1000 hours. With IRS there is just nothing to go wrong. No electric motors, no servos, no gyro bearings, no heavy electrical engineering. The MTBF has become virtually irrelevant. People bang on about the increased accuracy of IRS, but they miss the point. Sure, it is more accurate but, far more important, it's more reliable and cheaper. It hardly ever fails.
3) The vastly increased power of modern computing. In the old days you had to have a gyro-stabilised platform to keep the accelerometers level. The platform stayed level and north-orientated whilst the aircraft manoeuvred round it. This was expensive and it tended to fail every so often. Now, with real-time computing you bolt the accelerometers and lasers onto the airframe and they manoeuvre with the aircraft. However, the computer knows which way is north and which way is up (or down) and resolves these accelerations by matrix algebra and axis transformation into north, east, and down.
There are lots of other differences as well, but that's enough to be going on with.