Not really, it’s the difference between retirement at rank, and temporary appointment. To this day, the DoD official policy permits the use of the rank after retirement. https://kb.defense.gov/PublicQueries/publicQuestions/FaqsAnswer.jsp?Subject=Use%20of%20Military%20Rank%20by%20Retired%20Service%20Members&FaqID=202&nHit=0

Retirement is a bit different to separation. If someone separates or resigns the commission, one has relinquished the right to the rank, or, if you wish to consider it another way, the rank is taken back from them when they separate. However, if someone has retired at a rank, it’s more considered that the military has no more need for you and you are put out to pasture, but you are not removed before your time.

The details of retaining titles are interesting, but the main question from the OP was: why was a President ever addressed as “General” if the former is also Commander-In-Chief, and inherently higher ranking?

If I could ask a follow-up question, were other retired generals turned Presidents commonly referred to as "General" during their tenure as President? (Such as Washington or Grant)