The 7 Plus is a much more refined iPhone internally over the 6s Plus. Physically they are nearly identical with the exception of the deleted 3.5 Jack And with the dual camera.
The 7 Plus has outstanding sound quality with the stereo speakers and additionally has two hours of battery life over the 6s Plus, which would be appreciated with your gaming needs you mentioned.
Being you have children, the dual camera is nice to have with the 10x optical Zoom for long distance shots and Portrait Mode is an interesting camera feature exclusive to the 7 Plus.
The 7 Plus seems more responsive over the 6s Plus with the A10 Fusion processor and 3 GB of Ram.
The only difference with the 128/256GB models is the write speed makes the iPhone reportedly quicker over the smaller storage models. But other forum members have mentioned it's not truly all that noticeable.
The Haptic home button is interesting, but isn't truly an advantage over the 6s Plus home button. The 7 Plus also has a IP67 Water resistance rating.