They don't contradict. Flying Stars basically tells you the dynamic aspects of Qi affecting your home. In this case, you are looking at the annual cycle. 8 Mansions on the other hand is a more static view of the Qi of your home. In general, your NE sector is good for most of the time, except for this one year. As you progress to higher levels of Feng Shui, there'll be more layers of analysis. Different theories don't contradict, they just give you a deeper understanding of the Qi map of your home.
Warm Regards, JY