Difference in meaning between “elderly” and "old"?

Solution 1:

Is there more of a negative association to the word “elderly” than to the word "old"?

Possibly it is the reverse. I would normally call myself "old", and leave the usage of "elderly" to my grandchildren when they are talking about me within earshot. I am pretty sure they use other and more colorful language when I am not around.

Elderly is seen as a polite expression, especially when talking about a group.

Cambridge online

Ex.

The elderly are more susceptible to the virus.

...but usually not...

Old people are more susceptible to the virus.

A famous use of the word can be seen in LoTR:

Frodo: "We're friends of Gandalf the Grey. Can you tell him we've arrived?"

Butterbur: "Gandalf? Gandalf? Oh yes! I remember: elderly chap, big gray beard, pointy hat… Not seen him for six months."

...and of course Butterbur was being respectful, as any good publican would be.

Solution 2:

"Old" is an objective description, as a comparison with "young" - though there is no agreement on the age when one ceases be "young" and becomes "old".

"Elderly" is not necessarily either positive or negative, but it usually implies the idea of displaying signs of being "old".

For example someone still competing as a power-lifter at age 89, despite several medical procedures including triple heart bypass surgery, could certainly be described as "old" but not as "elderly." (Ref: https://www.bbc.co.uk/news/av/uk-england-beds-bucks-herts-51825800/bionic-powerlifter-ted-brown-still-competing-at-almost-90).

On the other hand a 50-year-old with poor physical mobility and declining mental capability might well be described "elderly".

For example, see the UK road sign warning of "blind", "disabled", or "elderly" pedestrians crossing the road: https://www.road-signs.uk.com/Road_Signs/Elderly_People/82.html - though some pensioners' organizations have complained that this is pejorative.

Solution 3:

The succession of old by elderly, and now by senior or older, is just one example of the euphemism treadmill identified by Steven Pinker. Perhaps we are not far from a complete cycle of the treadmill, when (for a while at least) the unpretentious old will be back in favour.