I got a Google Home for my wife for Christmas. She uses a Pixel, so I figured the Google Home would just work better in terms of integration moving forward.
IMO, unless you plan to use it for music for a majority of the time, get the Home Mini. The only difference between the two devices if the size/quality of the speaker, and the touch pad at the top of the Home. The Home also comes with a physical mute button that lets you stop the device from being in "always listening" mode. Not sure if the mini has that button.
Our device is in the kitchen, which is easily the hub at our place. Few friends come over and we inevitably end up hanging out in the kitchen, so the Home is great for playing music there. I always like coming downstairs during the week and saying "tell me about my day", and it will rattle off the time, day's forecast, my appointments and the news headlines all within 5-10 minutes, while I'm making breakfast. It can also read out recipes to you one step at a time.
Initially, the plan was to get two Home minis (one for basement, and one for second floor), but the basement TV now has google assistant in it, so we linked it to the Home and voila. (That hasn't really been of much use yet though. We wouldn't screen anything on the TV downstairs when we are upstairs).
I'd recommend it, but it really depends on your lifestyle.