If you're booking a studio, the rooms are essentially the same between both. The big differences come when you're booking a 1BR+.
Kidani has an amazing splash pad/pool area for little kids. If your kids are in the 8 and under group, they'd love it. The main pool at Kidani is a lot smaller than Jambo House's pool but they both have slides. The Jambo House pool also houses flamingos that you can watch.
Jambo House has the only quick service restaurant so I consider it a lot more convenient. Kidani has Sanaa, and Jambo House has Jiko, Boma, and The Mara (quick service). We ate at all of the restaurants except for Jiko and they were all excellent! Kidani does have a little gift shop with a few grab and go food items, but no hot meals like The Mara.
Kidani's lobby is pretty but very subdued compared to Jambo House's lobby which is huge, grand, and gorgeous. Bus service always picks up first at Kidani, then picks up at Jambo House. It's the same way on the return trip as well -- Kidani first, then Jambo House. Both resorts offer educational activities for kids so if you might be interested in some of those offerings, check the activity schedule. There were a lot of neat choices!
We stayed 9 nights at Jambo House (
DVC points as well) and loved it. The convenience to The Mara was awesome and we ate there a lot. We walked over to Kidani twice (once to eat at Sanaa, and once to check out the pool) and it's about an 8-10 minute casual walk. If you plan to eat at The Mara several times, I would definitely choose to stay at Jambo House. Both sides of the resort are super quiet and relaxing. I can't imagine staying anywhere else now.