The word eternal chains makes no sense because eternal has no beginning and no ending.
There are a couple other translations that use the word unseen and imperceptible that makes more sense when translating aidiois.
Translations from Aramaic
Aramaic Bible in plain English
and those angles who kept not their Principality, but abandoned their own way of life, he has kept to the great Day of Judgment in unseen chains under darkness,
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Jude 1 Peshitta Holy Bible Translated
6 and those angels who kept not their principality, but abandoned their own way of life, he has kept to the great day of judgment in unseen chains under darkness
Besides, messengers who kept not their own sovereignty, But leave their own habitation, He has kept in imperceptible bonds under gloom for the judgment of the great day.
In other words these messengers are imprisoned awaiting their judgment on the great day.
We know angels don't die so it is invisible to us to see how they are being held, unlike Sodom and Gomorrah, we can actually see the results. We are told they are not getting away with what they did. They also are not allowed to do anything else at this time either.
These bonds are definitely not eternal. It is compared to the judgment of Sodom and Gomorrah and the cities around them.
"The fire is out but the destruction is still apparent. And this way the cities are experiencing the justice of the eonian fire. The fire has long ceased but its effects will remain and testify to God's judgment until the close of this eon, after which Sodom shall return to her former estate. Ezekiel 16:53-56
Looking at these verses below shows that the judgment against Sodom and Gomorrah is for a certain amount of time. It is not eternal. This would also imply that the judgment of the messengers will be for a certain time as well. αἰωνίου (aiōniou
experiencing the justice of eonian fire. Jude 7 Concordant
an example, of fire age-during, justice suffering. Jude 7 YLT
αἰωνίου (aiōniou) —
Cognate: 166 aiṓnios (an adjective, derived from 165 /aiṓn ("an age, having a particular character and quality") – properly, "age-like" ("like-an-age"), i.e. an "age-characteristic" (the quality describing a particular age);
Here is the example what that will look like as given in the following verse.
53But I will restore Sodom and her daughters from captivity,d as well as Samaria and her daughters. And I will restore you along with them. 54So you will bear your disgrace and be ashamed of all you did to comfort them.
55And your sisters, Sodom with her daughters and Samaria with her daughters, will return to their former state. You and your daughters will also return to your former state. Ezekiel 16:53-55
He has kept in imperceptible bonds under gloom for the judgment of a great day. As Sodom and Gomorrah and the cities about them in like manner to these, committing ultra-prostitution and coming away after other flesh, are lying before us as a specimen, experiencing the justice of eonian fire. Jude 6,7 Concordant literal
I would imagine that the bonds that the messengers are chained in speak loudly to the celestial realm since that's where they left their domain.
OP'S question is,
"Jude uses "aidiois" and "aioniou" in close proximity to each other, perhaps suggesting that they have for him different meanings or connotations.
What contrasts might there be between them?"
This whole chapter has to do with severe judgements. One judgment we still see the results on the the earth. The judgment for the angels are invisible to us and we cannot for see what they look like. Rest assured God wants us to know that they are being held in prison awaiting their judgment.
The contrast are the different judgments, even though the angels and men did the same thing basically on earth. Both have different realms were their justice will be served out.
The thing they both have in common is that their judgments will be for a certain amount of time or for an age.