At its root this was originally a British vs. American English thing. However, the spelling of “disc” with a C has been used specifically as the trade name of various brands including both the throwable and optical media varieties, which have since become genericized trademarks.

For the optical media side of things, the name was coined by Phillips while they were consorting with Sony to develop the standard and named it the “Compact Disc” to compliment their already existing “Compact Cassette” product. They developed an official logo for the format which spelled it “disc.” That’s been with us ever since.

Volumes of computer storage are now colloquially referred to as “disks” because A) a significant majority of the early computer development milieu in general happened in America where we, or at least IBM, spell it with a K, and B) for a very long time, that’s exactly what they were. Tape and magnetic core memory and wire loop memory were all early developments that ultimately gave way to the longstanding popularity of magnetic platter/disk fixed storage… With some exception granted to tape, which hung around for a very long time but definitely was not a random access storage medium suitable for general purpose applications whereas disks were. It’s probably pure happenstance that the dominant non-fixed computer storage media also wound up being disk shaped, namely the various sizes and types of floppy disks. Computers handle linear tape based storage and random access disk based storage very differently, and nowadays random access permanent storage still has the “disk” moniker stuck to it even though it’s now likely to be solid state.

As a generalized descriptor of a flat circular object, either “disk” or “disc” is appropriate but which is preferred seems to be largely depending on which continent you’re from. The root of the word is indeed the Greek “discus,” as in the object yeeted across the playing field by Olympic contestants.