For me it's the ease of handholding and the size - it just makes it more enjoyable to use for most tasks, and makes it much more portable. While I don't have a full size iPad, I have a 10" Android tablet that I've owned for the past year and a half. (well, it's an HP Touchpad running Android). It's roughly the size and weight of an iPad 1. I know the more recent ones are lighter and thinner, but the overall dimensions are similar.
Last night I took my daughter to swim lessons. I usually browse my iPhone when there and such. I would NEVER contemplate bringing my Touchpad, nor a full size iPad with me. For one, I'd need to carry it separately, and there's nothing to rest it on for the 40 minutes of sitting, so I'd be holding it awkwardly that whole time. It would just feel too big to bring as a diversionary device. However, I didn't give a second thought to bringing my mini. It fits in my inside jacket pocket, so I didn't need to carry it, and it's small enough to be somewhat discreet while providing a superior experience to my phone. I pulled out my iPhone, tethered my mini, and used the mini the whole time. It was great. I'd never do that with a 10" tablet. I COULD, but it's more trouble than it's worth.