Dumbledore wasn't killed by the ring being a Horcrux; he was killed by the curse on it.
The curse was in addition to, and separate from, the Horcrux nature of the ring. It was placed upon the ring by Voldemort as one of his precautions to protect his Horcruxes.
"The ring, Harry. Marvolo's ring. And a terrible curse there was upon it too."
-- Dumbledore, HP and the Half-Blood Prince
"Why," said Snape, without preamble, "why did you put on that ring? It carries a curse, surely you realised that. Why even touch it? [...] That ring carried a curse of extraordinary power, to contain it is all we can hope for; I have trapped the curse in one hand for the time being —"
-- Snape, HP and the Deathly Hallows
Whereas Arthur Weasley was bitten by a dirty great snake. His injuries were nasty and probably life-threatening, but they were purely physical: there was no magical curse acting on him. Just being a Horcrux doesn't make an object more or less lethal (just perhaps more evil).
Clearly Voldemort didn't put a similar curse on all his Horcruxes, since Harry, Ron, and Hermione carried the locket around for weeks without suffering any lasting ill-effects. (It did have a psychological effect on them due to the evil magic surrounding it, but that's a lot less serious than the curse that killed one of the greatest wizards of the age.) Presumably he thought they were already sufficiently protected: the diadem in the room he thought nobody else knew about, the locket hidden in the cave with multiple protections, etc.