I'm not sure it does, myself. Look at the amount you can earn from the Club World Cup and that currently has very little prestige.
I think what the competition qualifies you for, Champions League for Europa League, for instance, makes it much more important to the club financially, but I can still see why a fan would think the FA Cup is 'bigger' than the Europa League. It's been fucking hard for us to win. If we were in the Europa League every year for the next 5 years, I'd back us in that competition much more than I would the FA Cup. It all depends if financial importance is the main qualifier for 'bigger'.