I skimmed the other answers, and didn't notice this but just in case it hasn't been said: adjoining is side by side, without a door between rooms. Connecting means a door between rooms. With little kids, I would want connecting.
We were told in May that connecting was "guaranteed" for our party at Pop. I was pretty sure it wasn't based on the fact that we had several adults in our party. But our reservation did say connecting, and we did get connecting. We have always got our request for connecting at Pop.
We have not at deluxes. For example, on our May trip we did get connecting at Pop but when we moved to WL three days later, we did not. I had told them connecting was very, very important to us as DD had three small kids and was traveling without her husband, and expecting a fourth baby. We anticipated she would tire easily and I wanted to be connecting so I could help her. We did not get it. We did not raise a fuss, as we knew it was only a request and not guaranteed. Point is, it seem's like it's a lot easier to do at the values.