Welcome to the forum newcamper!
Both are very similar.
Cost: Green mountains would be cheaper, as it is run by QNPWS, and has all the facilities though a bit worn. Sites can be a bit small if you have a large tent. Binna Burra is privately and you pay more, though you could get a small caravan in there.
Things to do with kids- Again probalby Green Mountain as there some short walks and the bird feeding.
Views - Binna Burra is problaby better, but again is more exposed to the winds.
Binna Burra has alot of shade int eh camping area which is great in summer but perhaps not in winter. Green Mountains is a little more open to the sun.
I think the mobile reception is good at both places. thats if you need it of course.
Binna Burra probalby has a better road up than Green Mtns, and is probalby a bit shorter.
If you want to avoid the coaches and the Casey Stoner wannabees on the way up to Green Mountain and you have a 4wd you can go via Duck Creek Road, which is quite a pleasant drive.
Don't know of that helps.