1) If the car has a 'modified' MacPherson strut system, with a more inboard spring not mounted over the damper, the spring has to be very large and heavy, as well as binding (buldging outward on compression due to being closer to the pivot point). Coil overs move the spring outward and over the damper where they have more linear movement and more leverage and can achieve the same wheel rate with a much lower spring rate and smaller lighter spring.
2) Adjustable spring perch with a simple spanner wrench for ride height adjustment.
3) Solid steel spherical bearing in the top plate eliminates flex, binding, creaking noises, suspension bounce, and wear and tear in the strut mount which is usually nothing more than a rubber donut.
4) Damper itself can be rebound/compression adjustable regardless of having a complete coilover replacement. Or in some cases someone simply wants a stiffer setup and the only means available for that make/model is with a complete aftermarket coilover conversion.
5) Upper mount is mounted in adjustable plates that allow quick and easy independant camber and caster adjustments.
6) Weight reduction, many coil over kits in addition to the spring weight previously described, focus on weight reduction using aluminum bodies, etc.
The term "coilover" was originally coined as it related to replacing stock suspensions that had a larger heavier inboard spring not over the strut, or even leaf springs. Most new cars today are already "coilover" stock, but replacement setups that are chosen for the other benefits are still called "coilover conversion kits".
So OPs example of an M3 which already has coilover struts from the factory, might have the entire coilover setup replaced with an aftermarket one that is adjustable or stiffer or has better mounts, but it was already "coilover" before it was replaced, so it's not really being "converted".
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