It really doesn't amount to much, you battle her instead of Florin, instead of going to Cassandra's office, you instead go to her gym with Erin who doesn't know the truth and end up in a Team Bladestar rally held by her, and Erin pretends to join to have a chance to get at Flora.
Additionally, instead of the Grunts kidnapping Reina by accident, Kanon brings you to the desert as planned and you meet up with Ren later when Alexandra tells you to go to the Chasm to pick him up.
During chapter 13, the plot still goes the same way, you battle Team Bladestar on the way to the Pyramid, and Flora's boss dialouge will acknowledge you as a member of Bladestar and tells you she has no use for you. Aelita even expresses surprise that we are members of Bladestar but Flora explains to her it was blackmail. Also Florin is still held hostage by her.
As Zumi back in the AMA notes, Flora was never intending to play nice with us, despite being a competent battler. It's Flora's loss if she doesn't realize we are a Player Avatar.