Hi Denise,
Welcome to the CiC forums from me.
Unfortunately that is just about the worst kind of subject to judge image quality on.
At 800iso, on a camera that probably has the same sized sensor as your P&S, shooting jpg, at 1/64s exposure time with probably a bit of breeze moving the leaves - you are asking a lot.
Try a solid subject, with more luminance contrast, in brighter light (hence allowing a lower iso and higher shutter speed) and properly exposed.
I am not familiar with either camera, but apart from a wider zoom range and probably manual controls, I would guess that in quality terms, there's not
that much to differentiate between them. However, the Fuji will hopefully allow you to make informed decisions to get the most from it and we can certainly help with that. UPDATE as you can see above already
Have you got a less challenging (photo) subject we could help with?
All the best,