Difficult one. Bigger rims in general look better(IMHO not on minis) and house bigger brakes. However, lower profile tyres transmit more road noise, they are heavier unsprung weight so the car handles worse, larger moment of inertia so can affect acceleration and will generally be wider, which reduces economy and isn't neccessarily grippier round the corners. They can also increase torque steer and wearing in the bearings etc
Smaller tyres have smaller brakes but:
Cheaper tyres
Cheaper wheels
Not as low profile so not as easy to curb(as pgarner said)
Better acceleration
Better handling(on the whole)
Also, if you have standard suspension, it will be designed for a certain diameter tyre. Mini's handle the best on 10" wheels for example. So, unless you change the suspension for the wheels you use, it will probs make it worse.
Dave