Sack, Fall, Conquer, they pretty much amount to the same thing. Maybe the article should have used the term "Sack", but then if you google around it's also known as the "Fall" or the "Siege".
A sack in this case would be different because in the end it was still controlled by Christiandom, as opposed to the Fall when it switched to being controlled by the Ottomans. I think these small details do matter in this case, but maybe its just me.
Before the 1204 sacking it was controlled by Greek speaking Eastern Orthodox christianity and afterwards it was rued by western catholic christians with a Latin mass and a pretty fundamentally different culture. Its known as the Latin Empire and lasted about 70 years before one of the surviving remnant parts of the Byzantine empire managed to recapture the city. So while it was not The Fall Of Constantinople it was certainly A fall of Constantinople.
Having checked various dictionaries, I'm certain it doesn't matter who, religion, ethnicity or race-wise, is on the dispensing or receiving end of a sacking, fall or conquer, they're pretty much the same. That said, sacking seems to be a specific case that involves looting and running off with the good stuff, i.e. plundering, which is one of my favourite words.
I guess the article makes no distinction between a Sacking and a Fall.