Hello, and welcome to the Apple Communities!
I suspect the difference compared to your "old macbook" is that FCE is not supported on Mountain Lion. FCE was discontinued even before Lion was released, and isn't even supported on Lion much less Mountain Lion.
How did you even get FCE installed on a new MBP with Mountain Lion?
One other thing that comes to mind is that you need the FCE 4.0.1 update in order to effectively use AVCHD video ... do you have that update installed?
Another thing to look into is whether you recorded in 1080p or 1080i mode. FCE only supports 1080i mode. If I recall correctly, iMovie will capture both 1080p and 1080i.
Exactly what version of OS X do you have installed on your "old macbook"?
To install, just put the CD in the drive, lol - and I did do the 4.0.1 update. The video is in 1080i mode. As a test, I could load it into FCE on my old macbook which I still have which has 10.5.8.
So far, FCE seems otherwise to perform as usual on my new macbook Pro although I haven't tried to export a video yet.