I'd say there is a difference. When managers downplay their chances of winning the league, whether it was Mourinho at Chelsea or Ranieri last season who kept insisting they wouldn't win until almost the end, it's mainly to take the media spotlight and pressure off the team. To keep them from getting complacent even if they are downright favorites. I don't think that would dampen the mood at the club. Now I am saying that this would necessarily happen at Sunderland after Moyes's comments. Indeed, it might be his way of telling his players that they were in a dogfight every single match. But it could also lead to a few heads dropping. Which is not want you would want 2 games into the season.
Advocaat did keep saying that he would leave if Sunderland kept losing. But he didn't say this after his second game in charge afaik. Just to qualify, I am not objecting to what he said, per se. If this had been December or January, I don't think his comments would have been picked on. Admittedly, my personal dislike of him keeps me from reading his comments in a completely objective manner.