I have the iPad 3 and when the mini came out I had to get one. Since getting the mini I have rarely used the iPad 3. I just find the iPad mini easier to work with and a lot easier to carry around. The smaller size fits in my backpack or messenger bag better and the lighter weight makes it a joy to carry around. I never thought that the iPad 3 was heavy by any means, but when compared to the mini, the iPad 3 seems as heavy as a brick.
As for what I use it for, well aside from using it to read books via the Kindle app or iBooks, there is surfing the web, checking email, and playing a few games. I also use Numbers to keep track on a few spreadsheets, I use Pages to write a few letters with and I use PocketMoney to keep track of my finances. I am also user of CraftStudio and love it to create greetings with it. When I got the iPad mini I got a 64GB version and love the extra space where I can put a few movies on it when hitting the road. My iPad 3 was just a 16GB version so that wasn't always possible to do. Since I have a lot more space on this iPad I have also started creating some photo books in the iPhoto app. I take a lot of pictures with my iPhone 5 and with photo stream love that I can veiw a larger image a few seconds later on the iPad.
I have read a lot of negativity on the mini because it's not retina, but in all honesty I don't miss retina on the mini. The screen itself is pretty amazing for a device the at size. If a retina mini comes out that doesn't compromise the size and weight of the mini as well as the battery life then I will more than likely upgrade at that time. Until then the iPad mini will be my iPad of choice. I am even considering selling my iPad 3 because in all honesty it is just sitting there collecting dust at this point.