Disclaimer: I work for Microsoft on the Roslyn team.
CodeDom is a precursor to Roslyn, but is only marginally related. Essentially, CodeDom is a simple and (somewhat) langage agnostic way to generate code that was added in .NET 1.0 to support designers (a la WinForms). Because CodeDom was an attempt at providing a unified model that can generate code in C#, VB, and other languages, it lacks high fidelity with any of the languages that it supports (that's why you can't create a switch statement with CodeDom). CSharpCodeProvider.CompileAssemblyFromSource is simply a wrapper around executing csc.exe.
Roslyn is a completely different animal. It is a rewrite of both the C# and VB compilers from the ground up using managed code -- C# in C# and VB in VB (the versions of csc.exe and vbc.exe that ship today are written in native code). The advantage of building them in managed code is that users can reference the real compilers as libraries from .NET applications (no wrappers needed).
While building each component of the compiler pipeline, we've exposed public APIs on top:
- Parser -> Syntax Tree API
- Symbol Table/Metadata Import -> Symbol API
- Binder -> Binding and Flow Analysis APIs
- IL Emitter -> Emit API
Roslyn can be used as a sophisticated C# and VB source code generator, but that's where the similarity to CodeDom ends. The Roslyn Compiler APIs can be used to parse code, perform semantic analysis, compile and evaluate code dynamically, etc.
In addition to the compilers, the Roslyn team is also rebuilding the Visual Studio C# and VB IDE features on top of the public compiler APIs. So, the compiler APIs are rich enough to build the Visual Studio design-time tools, like IntelliSense and the Extract Method refactoring. Also, at layers above the compiler, Roslyn offers services for higher-level analysis or data transformation. For example, there are services for formatting code using the C# and VB formatting rules, or finding all references to a particular symbol within a solution.
Really, there isn't just one special benefit of Roslyn over CodeDom. Where CodeDom filled a very specific code generation need, Roslyn is tackling the entire language tooling space by providing a framework to allow you to build just about any sort of C# or VB language tool you can think of.