The big difference is the iso setting I think. The z850 will do iso 800 and iso 1600, but the noise reduction algorithm in the camera is a joke--they are really nasty pix when enlarged at all. If you compare the z850 iso 50 shots to, for example, F30 iso 50 shots, the z stacks up well--probably better. However, as noted in posts here, to use iso 50 or other low isos in low light or at night without flash, you've got to go to long shutter speeds. With a tripod or brace, that may be fine. However, since the F30 has pretty acceptable grain even in iso 800 and iso 1600, it is capable of taking similar low light shots at shutter speeds that can be (carefully) hand held. That's the difference. You can't do the low light hand-held shots with the z unless you put up with truly awful noise/grain. On the other hand, I don't believe you can manually set the shutter on the F30, so you probably can't get as good low light tripod pictures at lower isos no matter what.
There is a post here somewhere about using custom bestshots to get iso 800, 1600, and even 3200 from the z without the built in noise processing. I've tried this, and it really is amazing--you can get acceptable grain even at iso 1600. Using this method, I think the z850 is almost equal to the F30 in getting low light hand held pictures...but it's a bit of a hassle to set this up. The poster had BS jpe files you could use, but as I recall, he used a white balance setting on the iso 1600 and 3200 that made the pix orangeish. I modified them to get rid of that setting, and they work well that way. It's not too much trouble to use BS to shoot in low light, but it's a shame Casio didn't have better (or perhaps just less) noise processing to begin with.
Interestingly, the s770 has an iso 800 capability that is very good. I've had a lot of fun with low light hand-held shots (no flash) with it, and that quelled my interest in trying the F30. I'm sure the F30 is better that way, but the s770 is good enough, and has other advantages (slimness, sd card, ease of use etc).