While there is an overlap of skills across these areas, they focus on different aspects. UX design is focused on a single touch point in a customer journey. Most people are familiar with this touch point being a single website or app.
Service design considers the fundamentals of what is being offered to the customer and does it from the perspective of multiple actors, such as the customer and the service provider on an end-to-end basis. Service Design asks what problem does this solve (for the entire service).