Tough decision. The 70-200 is significantly sharper and has a faster f-number in the overlapping range, but the 70-300 has I.S. and an extra 100mm reach (which will give better pictures than cropping the 70-200). One of the first things to consider is what time values (shutter speeds) you'll need. Wildlife sometimes holds very still, but often you're at least 1/250 or 1/500 to start with. In the latter case, I.S. probably wont help much. For kids I don't usually ever go below 1/125, even if they're trying to hold still. But if they're running around and playing, I'm back up to 1/250 or 1/500 again. So for those two, I.S. may not be much help. But for other outdoor photos, such as landscapes or portraits of adults (who I can usually shoot at 1/60), the I.S. will come in very handy.

Then there's the extra 100mm reach. I would be thinking about how often I would use that, and how important those shots are. If wildlife was the most important consideration, you definitely want that 100mm. But if kids are the primary purpose, and you can cover most of the needed range with 70-200, then stick with the L.

Hope that helps.