Differences between UX processes for a new client and a redesign

I was asked a question recently about the differences in the UX process when starting with a new website from scratch, compared to the UX process of a redesign for an already existing website. I said that the processes are largely the same but with a existing client you have the benefit of having current statistical analysis to help inform your decisions. But it had me thinking if this is the right approach and is there more to this? What are peoples thoughts on this?