Why should an organization adopt the unified 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?
RUP was very much en vogue about 10 years ago, but - as @Developer Art commented - it isn't talked about that much lately, parallel to the advent of Agile methods. Frankly, the latter are way simpler, practical and straightforward than RUP. And you can use UML regardless of the methodology you pick.
For me the only reason to use RUP would be if the client/sponsor/management absolutely insisted on doing so. And even then, RUP can - and should - be tailored to the specific project/team, so I tried to keep it as simple as possible :-)
One "advantage" (it's in quotes for a reason) of RUP is that if you follow the RUP conventions properly, then two different developers will (mostly) arrive at the same OO Design for a system.
This is a Project Managers wet dream, it takes the Inspiration/Art/Flair/Joy out of OOD, and makes it so you can point pretty much any developer at any task and they can pick up the design logic and get up and running quickly. It also makes estimating pretty straight forward e.g. This use case suggest 6 screens, which is 6 sets of Module Controller Views, which will each take approx 3 days dev etc etc.
However it can lead to dreadful and clunky designs, exactly because it leads your design down a predictable path, all the rules were followed for everything. i.e. it can lead to Design by committee type designs.