I would expect that the rubber bands from the 80s were very probably vulcanised latex (natural rubber) and the bands from the noughties could be a synthetic material or latex.
All polymers suffer from oxidation, which is why vehicle tyres (typically made of styrene-butadiene rubbers these days) are coated in an impervious protective layer for storage. Motorcyclists are aware of the dangers of this preservative. New tyres on a moto are generally treated with caution for the first 50-100kms until the tyres are "scrubbed in".
Apparently latex is less susceptible to perishing via oxidation when it has been correctly vulcanised:
http://www.newworldencyclopedia.org/entry/Rubber#Reason_for_vulcanizingIn either case, UV light is a killer. So, if your old papers were kept in the dark and the new ones on a shelf in the sun then that would explain the performance difference.
So, I'd expect that latex will perish over time if you wait long enough. However, you might be witnessing a better production standard dating back to the 80s.