This is going to probably upset both sides of this hotly contested debate, but with the exception of cold weather climates, I could frankly care less. Despite my own refusal to worry too much about this subject, there are definite advantages to each method.
Robust rootstock allows roses to produce thicker canes earlier in the rose's life.
Own root roses are plants which are grown on the rootstock which is naturally part of that specific variety's genetic makeup. Grafted roses are budded onto rootstock chosen for vigor and health. When you receive your rose, check the area of the plant where the roots extend from the canes. Smaller nurseries or those that specialize in antique or old roses tend to produce roses on their own roots. The chief reason roses are produced on their own roots is that they are much more likely to survive very tough winters.