Final Cut Pro has notorious problems interpreting Alpha types (the way transparency is stored in the video file).
Select the clip in FCP's timeline and go to Modify > Alpha type. I would start by choosing "black".
Now, I am not sure FCP can handle premultiplied alphas matted against something other than black (suppose you original Background Color in AE was something else)
So, if changing the alpha type to black doesn't help, try rendering again from AE but this time make sure the Alpha is set in the Output Module as "straight". Then in FCP, select this clip and in Modify > Alpha type, select "straight".
If it's not this, there could be a field order issue. But check this first.
It turns out that I made a few mistakes. The footage is 1280 x 720 at 29.97 FPS. The animations I made were 720 x 480. I didn't realize this until I put them on my timeline and notice the wireframe outline for the animation was smaller than the footage. Although it is kinda weird as when I check the footage properties in Final Cut it shows that the size is 720 x 480. Strange. And the biggest mistake was that the preview window was not displaying quality due to system limitations. After rendering the animation and resulting composite looks a lot better. Although I the edges are not as crisp as I wish they were it is passable for now.
Here's the link to the video. I think it looks okay although there's obviously an interlacing issue.