It's my understanding that certain turtles don't exhibit telomere shortening, and are effectively 'immortal' as well in that they don't age.
I think that neither really makes them less 'immortal' in the sense that this article means.
Though I can see the argument that errors in cell replication leading to cancerous growths could be counted as a form of natural life span limitation, I would be surprised if it didn't also happen to this jellyfish.
That the telomere doesn't shorten basically means the DNA doesn't "age"; cells don't become more likely to mutate after X duplications begin to damage the code, like ours do.
But, sometimes these mechanisms fail and a cell survives with mutated DNA. If the right genes are mutated, then it grows uncontrolled and you've got cancer.
In order to get immortality, these repair mechanisms would have to be as robust after 1000 years as they are after 20 years. In humans, the repair mechanisms tend to degrade as you age, which is why, the theory goes, people are more likely to get cancer as they get older. It's not that there are more errors then, but that they're not corrected as well.