That's my question exactly. TFA doesn't really answer it clearly. The only similarity mentioned is that the giant virus has a regular geometric shape, presumably due to capsid proteins as in other virus. In addition, other commenters, are saying that they are parasitic they just require much less of the host machinery to reproduce than a classical virus.
Also, the article mentions that there is some debates as to whether these giant viruses represent an new kingdom of life, which if it were true would imply that they aren't viruses, but members of some new classification.