The cassette continued on as long as it did because of the convenience of the Walkman; portable CD players just were not very practical due to size and skipping issues, though the latter issue was mostly solved around 2000 or so. Also, many cars still were equipped with cassette players well into the 2000s, and it was still found as a feature on boomboxes just about as long, if not longer. Personally, I remember seeing ads for CD players at decent affordable prices starting in the late '80s/early '90s, and I personally got my very first CD player (a Sony boombox) as a Christmas present in 1992. It had a cassette recorder built into it.

The format which the CD killed off was the LP record. The CD made its debut in 1982, but CD players were too expensive for most consumers, and the discs themselves had sound quality issues early on thanks largely to recording engineers who were used to mastering LPs. Once those issues were sorted out, sales climbed dramatically, as sales of LP records started to fall. By 1992, the 33RPM LP record was all but dead at the retail level, though the 45RPM 7" single continued on a little longer than that due to a lack of a suitable CD equivalent (3" mini-CDs were attempted, but weren't much cheaper, and were awkward for some CD players to handle) and due to their use in jukeboxes. By 1996, CD players were commonplace in the consumer market, so the original Sony PlayStation definitely didn't have the same effect on the audio CD that the PlayStation 2 would have on movie DVDs when it was released.
-Adam