If I understand it correctly, functor gives you a new set of functions from a type you provide
More generally, functors map modules to modules. Your Set example maps a module adhering to theORDERED_TYPE signature to a module implementing a set. The ORDERED_TYPE signature requires a type and a comparison function. Therefore, your C# is not equivalent because it parameterizes the set only over the type and not over the comparison function. So your C# can only implement one set type for each element type whereas the functor can implement many set modules for each element module, e.g. in ascending and descending order.
Even if functors are just generics-for-namespace, what's the significant advantage of that approach?
One major advantage of a higher-order module system is the ability to gradually refine interfaces. In OOP, everything is public or private (or sometimes protected or internal etc.). With modules, you can gradually refine module signatures at will giving more public access closer to the internals of a module and abstracting more and more of it away as you get further from that part of the code. I find that to be a considerable benefit.
Two examples where higher-order module systems shine compared to OOP are parameterizing data structure implementations over each other and building extensible graph libraries. See the section on "Structural abstraction" in Chris Okasaki's PhD thesis for examples of data structures parameterized over other data structures, e.g. a functor that converts a queue into a catenable list. See OCaml's excellent ocamlgraph library and the paper Designing a Generic Graph Library using ML Functors for an example of extensible and reusable graph algorithms using functors.
Classes can also be used as ad-hoc namespaces using nested classes.
In C#, you cannot parameterize classes over other classes. In C++, you can do some things like inheriting from a class passed in via a template.