I would say the only real big ( or small, lol ) difference is that you can't ride a mini. Not as an adult at least. Some minis shouldn't even be rode by children do to the size of the mini.
The only other thing I can think of is the food. You use less hay to keep a mini healthy.
Other then that, minis can drive like big horses, get colic like big horses, and have the same health problems as a big horse.
Now with all that said, I am somewhat new to minis, so I am just stating what I have learned since having him. I did have bigger horses when growing up so I have some idea on them too!
I can tell you one thing that you can do with a mini that you can't with a big horse. Take it in your car!! LOL Well, I guess that is actually something you can do with a big horse now to. Now that they have those huge Dodge Vans! LOL
OH, one more thing, I think you can keep a mini on less land then a big horse. I have about an acre, with a 1/2 acre that my mini can run on. I think a big horse is to big for that small of property! Just my opinion though!
OH, one more again! It is great to have a mini!!!! I love mine to death!