I realise that last guy who took this viewpoint got lambasted but I don't consider Sunderland a 'big club' or necessarily bigger than us by virtue of them playing in England. They have decent attendances in relation to how shit they are, I agree. But really, they aren't a team that look like they can aspire to much. Aberdeen is an attractive job because there's a potential to play for trophies and play in Europe. Sunderland has more potential and exposure, i.e. the potential to triple your salary and play on the 'big stage' of the EPL. But I don't think it's a foregone thing that Sunderland are bigger than Aberdeen purely because they're rich and can afford decent facilities etc. We have a slightly more glamorous history, including European trophies, which, to your old school fan, would make us the bigger team. I am aware that in the modern era of MOTD, Twitter, SPORFBANTS accounts etc that your average joe has probably not even heard of us.
I would say they have potential to be a big club mainly because of their fanbase. Don't particularly dislike them, but they're a team that's been horrifically run for years (kind of like ourselves a few years ago) and need rebuilt from top to bottom. I digress, though.
The 'big club' debate is entirely arbitrary though, anyway.