Concrete Overlays , spray , stamp , ect. is what i have done in my business since 2002. So i can lend some insight and also a professional opinion if needed.
In the short tho:
Both materials are essentially the same thing. Both a "portland" based material, both have acrylics/resins for movement and bonding. Both will require re-sealing at certain times. Both have unlimited colors to choose from. The big difference between the two are basically the look you can achieve and also of course the cost of each. Stamps being the most expensive. I personally love the look of a stamped overlay done correctly as opposed to just a "spray" pattern. However the durability is fine with both but you mention having a lot of kids running around.... the stamp pattern is MUCH more slippery then the Spray. The safety of your kids and the neighbors kids, ect usually come into play when thinking about which direction to go.
As far as a contractor/ MAnufacturer i can help you with that as well. Obviously I have nothing in this cause im from PA but i have used alot of different materials over the year and can at least help you locate the proper things you will need and the proper people you nned to speak with .
Hope this starts helping a little bit.