I have the note 8.0 (i'm using it now to post this) and my 2nd daughter has the ipad mini. My other 2 daughters each have the ipad 2.
It is a matter of preference. Since i already have the S3 it was easy for me to start using the note 8.0. I do find it hard to use the ipad, but i do manage. My daughters and my son find it very easy to switch between the 2, so when they get tired playing their games they hijack my note 8.0.
The note 8.0 has some obvious advantages over the ipad mini. 1st, the SD card slot. I put all the apps in the internal memory and all the other stuff like mp3, movies, pics, etc., in the external SD card. If i need more storage i can just swap SD cards. Also all those files, apps and data, both internal and external, are downloadable and uploadable to a laptop or desktop. With an ipad you need to jailbreak. Google Appstore is also very easy to use. With ipad you need to put in your credit card info even if you only intend to use free apps while there is no need for credit cards with appstore free apps. And paid apps are cheap, average $1 each. Another thing, note 8.0 has a quad core, ipad mini has a dual core. So in the case you will be using an app that needs a quad core you can run it full speed with a note 8.0 but not on an ipad mini. Jelly bean (android 4.2.1) also has multitasking, allowing you to switch from 1 app to another without the need to stop the apps. You can't do that on an ipad (please correct me if i'm wrong coz last time i tried i can't).
btw, the note 3 is rumored to come out in october if you need something smaller. It is anywhere between 5.9 to 6.3 inches super amoled screen, has the newer 4.2.2 android version, and a 8-core processor.