Okay there really isn't even a comparison here...seriously! But since I did own a Cricut Mini I am gonna discuss it anyway *okie is wondering just how short this versus post will be*. I acquired the Cricut Mini on the second Cricut Cruise I went on and it was something I never considered buying.
I really didn't get the concept of the machine and maybe it was because I really loved my Expression and being able to cut 12x12. I know some people were excited about the size of the machine because they felt it would be easier to take to crops and such but if I remember correctly it was difficult for a lot of people due to the requirement of having to be connected to the internet to use it and not all venue's had internet capabilities.
Now I will say the Mini did cut great and it was more quiet than the Expressions when cutting. I know it even cut glitter paper quite well. The small size of the machine was nice and it was easy to store. After using it a few times I put it away and went back to my Expression. I ended up donating the Mini and the mats to a school teacher I know.
As for comparing it to the Explore, like I said earlier, there is really nothing to compare...EXPLORE WINS. I guess I can say the Explore wins because of the superior cutting abilities, the quietness of the machine while cutting, ability to cut 12x12 material, cutting a wider variety of materials and yada yada yada. And even though I hate to admit it the Cricut Design Space program is easier to use than the Cricut Craftroom program and so far I haven't had as much problem with the internet connection as I did with Cricut Craftroom but that maybe due to me only using Design Space for cutting and not designing.