In the more traditional projects that I've worked on, the project manager (and, on larger projects, there might be associate/deputy/assistant project managers should one person be unavailable) is the person responsible for communicating with the customer, receiving project health and status updates, determining scheduling and budgeting, managing the process, ensuring the team has what they need to complete tasks, and so on.
In Scrum, however, these responsibilities are split between the Product Owner and the ScrumMaster. The Product Owner is the voice of the customer. They interact directly with the customer, create user stories, organize and prioritize the product backlog, and other user/customer facing issues. The ScrumMaster handles the process, overseeing meetings (including estimation and planning), removing impediments, and monitoring the overall health of the project, making adjustments as needed.
I've read in multiple sources, including Wikipedia, that the role of ScrumMaster and Product Owner should be held by two different people. I've not only read about, but worked on successful "traditional" style projects where the activities of both were handled by a single individual. In fact, it makes more sense for one to three people to be responsible for handling project (including human resources/staffing) and process level tasks, as they often go hand-in-hand. Process changes have an impact on scheduling, budgeting, quality, and other project-level goals, and project changes have an impact on process.
Why does Scrum call for isolating these activities into two roles? What advantages does this actually provide? Has anyone been on a successful Scrum project where the Product Owner and ScrumMaster were the same individual?