Why adopt the UP process over others? What are the relative advantages? I know that it is closely coupled with UML but clearly this cannot be the only advantage? Why choose this approach over others?
I think it really depends against what process/methodology you compare. Without details, only general characteristics of UP can be mentioned. It is iterative incremental methodology with well described roles and activities, using modeling techniques in object oriented analysis and design. It is vertically (time) divided in phases and those in iterations and horizontally into groups of activities concerning different aspects of software development, such as requirements, analysis, design, testing deployment etc...