The Mini is often by the bedstead used for ebook reading in the evenings. The Air is the "workhorse" for web browsing (or "fun house" or "time waster").
The Mini is also perfect for photography-- right size to hold, right size to view. The Air is too big for that.
I also use the Mini for astronomy far more than the Air -- hence, the user name Saturn--again, the size is right for all sorts of astro apps. It is more intimate.
The Mini falls down, though, in that the text on many web pages, especially Apple's own (i.e., the forums), is too small and rotating doesn't enlarge it enough and pinch-zooming results in text one has to scroll back-and-forth on as it spills off the screen. The Air works much better for that, although, truth be told, Apple's support site is just NOT well matched for the iPad!
The big screen is also nice for readings news apps, weather sites, daily comics, and playing games with a partner where we can both look at the screen more easily. (Although we've found that still can strain the neck, so we end up sometimes bringing up the app, say a word puzzle, on two devices at the same time.)
The Air is also much better for streaming sports, shows.
We're using them, overall, probably equally, but for different purposes. We'd be hard pressed to have only 1 size in the house!
In fact, we've been living with two iPads and one Mini for a couple of years now.
I have the Air 2 and the mini 4. I prefer the Air for home use because of the larger screen. The mini has LTE and is the device that I take out of the home because of its portability. I use the Air a lot more, probably because the mini competes with my Kindle when I leave the house. If I had to give one up it would be the mini, although it is a great device and runs as well as the Air. My main complaint would be that the smaller print is hard for me to read on websites, but I'm sure that is age related.
I bought a Mini 4 last week and it's a bit too early to say. I still use both and mainly have been using the Air 2 for YouTube and other videos while the Mini 4 is more my browsing tablet.
I will probably end up keeping both. I had an iPad 2 that my girlfriend was using a lot to watch videos so I will probably end up giving that to my parents and keeping the Air 2 and Mini 4 to use between myself and the gf.
I also personally own a rMB and 6s Plus while my girlfriend has the regular 6.
I've got an iPad mini 4, while my father has the bigger iPad Air 2. Both tablets are incredible in use, but the screen size is definitely the biggest change between them. iPad Air 2 is best for typing a lot (until you connect an external keyboard to the mini of course), as well as for watching video. But iPad mini 4 can do everything the normal iPad can do, but it's easier to carry with you.
I personally prefer the iPad mini 4 because of that reason.
I have an Air 2 and Mini 4, just upgraded from a Mini 2. The Air 2 is wifi only and stays at home; the cellular Mini 4 lives in my bag and goes with me everywhere. Indispensable for travel. The Air 2 is just too big. At home, the Mini is just too small.