Second, I'm going to go out on a limb and say that neither are what you really want. If you want something that's an all-in-one solution, you need something in the middle.
Short version: the D5100 is great for stills of your kids but bad at video, and the FZ200 is bad at stills but good at video. Here's the long version:
The D5100 does autofocus during video, but only has a mono microphone and due to the large sensor, there's a lot of glass to move, so contrast detect autofocus performance is slow in video mode (or live view, for that matter). In video, it will not keep up with your kids running around. However, with stills, the phase detection system will outperform any compact camera at any price point, and let you capture your kids running around.
The FZ200 also uses the same AF system during video, but has a smaller lens with less to move, and uses micromotors that can focus faster with a contrast detect system. Additionally, the smaller sensor means that more is in focus than with the D5100. Unfortunately, when using the sensor's full resolution for stills, the contrast detection system can't keep up with moving children (it relies on the reduced resolution of video and the fact that you don't notice focus as much during video to make that work), so unless they sit still this is good for video but not photos.
As for the aperture, f/2.8 is nice, but the D5100 still easily wins in low light with its larger sensor - and you can always get an f/1.4 lens for the D5100, if you want, but you can't do that with the FZ200: what you get to start with is what you're stuck with forever.
Now for what I actually think you need:
If only there were a camera that had a sensor larger than a compact camera, but smaller than a DSLR, had interchangeable lenses so you could change it up to suit your needs, and would use phase detect autofocus to keep up with your kids running around in both still and video mode. Oh, wait, there is!
(Seriously, that's the thought process I go through whenever I recommend this to someone, it's an in between camera that's perfect for taking photos and videos of children, and is competent both stills and video).
The Nikon 1 system. B&H, if you're in the US, has two great bundles on the V1 right now for $300 and $700 - which is less than just the lenses in the two kits cost, and the $300 two lens kit is less than either will set you back (if you can't find the kits, PM me and I'll send you links). It will easily outperform the FZ200 in every metric, at a lower price, and the only place where the D5100 really beats it is with low light and depth of field isolation due to its larger sensor.
Anyway, that's my two cents! If you're stuck with the two though, get the D5100, it's much more camera for you to grow into and you can always stop down the aperture and shoot at f/16 to get more in focus in video if you need to.