To quote myself in the ASk the Apex thread:
One reason for F1 using 13" wheels is because the rim is heavier than rubber so a smaller wheel makes lighter wheel/tyre combo. This would help the designers with their weight distribution and, I would presume, allow them to make the suspension units lighter as they have to support less weight. A heavier wheel/tyre would also increase the stress on the drive shafts & gear box. I found a thread which estimated the increase in weight of going to 18" rims as being as much as 33KG - about 5% of the weight of an F1 car.