Crook, a former Tottenham Hotspur and Norwich City midfielder, stepped down with the club languishing in eighth place in the 10-team A-League with just two wins this season.
The 49-year-old Englishman had been on shaky ground after his team were thrashed 7-2 at Central Coast Mariners last week.
"It builds up. All this [media] has become a bit of a burden," he told reporters on Sunday.
"With Alessandro coming, the whole thing has become a bit bigger.
"It's become a bit of a weight on my shoulders, to be honest with you, and I haven't handled it that well."
Del Piero missed Sydney's loss to the Mariners due to a hamstring complaint but was welcomed back for the Victory match and the 38-year-old World Cup winner was influential in setting up both his team's goals.
"I don't regret taking it on... but I don't think I'm the right person," Crook added.
"It just became a really heavy burden, where it was getting to the stage where I just wasn't feeling well. And my health and my family are the two most important things."
Sydney were inaugural A-League champions in 2004-05 and have styled themselves as the fledgling league's glamour club.
"We have a lot of expectations," Sydney FC chief Tony Pignata said. "I can sense [the fans'] frustrations.
"We ask for their patience, I know some won't be patient."
