The first is that the Canaanite woman's daughter was possessed by a demon which Jesus then cures. Demons seem to possess people quite a bit in the New Testament. Were these actual demons or is demon possession equivalent to seizures? While I believe that demons are very real (if not how they are represented in art) and can affect our daily lives in ways both subtle and overt, there are many medical conditions with highly understandable and mundane sources that used to be attributed to the supernatural. One way or the other, Jesus heals her and the demon, whatever it might have been, troubles her no longer.
The second is that Jesus looks like a jerk at the start of this reading. The woman asks Jesus to heal her daughter, and Jesus equates her and her people to dogs. Now, the woman recovers quickly and, instead of lashing out at the insult, uses his term to justify her petition. Jesus then praises her faith and does as she asked. Today's homily says that this shows us why persistence in prayer is important, but I think it goes a step farther: We need to be prepared for Jesus to challenge us when we petition Him. Many people turn from God because they prayed for something and it wasn't given to them. God may have a different path in mind for us than what we plan.