I would add that both courses, while challenging, left me wanting to go right from the 18th green to the 1st tee. The only course I can think of that evoked that feeling even more than these two is National.
Of course, there are many differences.
As many pointed out, FH is much hillier (though still not a bad walk) and sandier looking.
You have to almost try to lose a ball at FH, while an offline drive on some holes at GCGC can result in a re-tee. (OK, maybe very off line...what can I say)
While almost British in feel, GCGC is probably the most unique course I've ever played. As you can see on this thread, almost every hole is filled with quirk and interest. I can't think of another course like it.
FH is also somewhat unique, though some of the front 9 holes are similar to other C&C courses. I do not say that as a bad thing - I think the front at FH is underrated simply because the back is so spectacular. I would put the final 5 holes at FH up against almost anywhere in the world.
As for non-golf stuff, I would not call FH "flashy" (the entrance is unmarked!). The clubhouse is certainly first rate, though very classically done, and I think will age well with time. It is also not as big as it looks in some pictures.
The clubhouse at GCGC may be the coolest golf-related place I've ever been. The mere fact that it exists makes me happy.
In the end, there is not much similar about the courses, other than the fact that a round at either place is guaranteed to be incredibly fun.
One final point: some people commented on how green these pictures look. When I played GCGC last year (in July or August) it did not look so lush, and played very F&F. I am guessing the presentation here has more to do with weather than anything.