The Significant Mistake:
Amazingly, the same one as Jar Jar, although in her defence she makes it first. Not only does she fail to see through Palpatine, she initiates the Vote of No Confidence in Senate leader Finis Valorum that allows Darth Sidious to rise to power.
Impact On The Saga:
The usual - Sith, war, Empire. With added personal baggage that Palpatine persuades her hubby to become a genocidal arsehole.
If The Force Had Been Strong With Her:
She'd have refused Palpatine as Valorum's replacement. After all, she's a Queen, she can do what the hell she likes.