As I understand it, Severitus is a specific challenge, like the MLC or the YLC. Specific rules and guidelines were laid out. Sevitus has Harry as Severus' biological/adoptive son, or just Severus as Harry's mentor; just so long as it's a father/son-type relationship, rather than Snarry, which is the name for their 'ship.

What's even more confusing is that I'm not sure if all the Harry-is-Severus'-son authors know the difference. I've never read the challenge, so I can't be sure, but as I understand it, my favorite fic in this genre follows the exact rules. The author calls it Severitus anyway.