There's an excellent article on the website of The Honourable Society of the Middle Temple regarding who can remove or revoke what in regard to royal titles, honorifics, peerages, etc.

The article was written as a response to the general public's clamor for Harry to lose his "titles" but is equally applicable Andrew.

Unfortunately, the color of the font used in the article is very faint, and older eyes will probably need to either highlight the text or do a copy and paste,

The reason why Andrew retains the rank of prince is simply because Charles has not issued LPs removing it, and Andrew has not agreed to cease using it.

We don't know what pressure Charles exerted to get the couple to agree to stop using the peerage titles.

In regard to the Garter and other gongs, Andrew hasn't worn any of them in public since Charles's coronation. Unless he's been spending his days writing myriad letters and signing them with a long list of post nominals, what has changed?

My guess is that Charles was able to apply enough pressure or offer enough of an incentive to get Andrew to agree to cease using the various titles.

Because Charles doesn't want to officially remove anything because it would set a precedent and Charles doesn't want to be pressured by the public and/or the media to mete out the same punishment to his own son.