Both products are wrappers above the C++ code.
Drawings.NET Classic is a set of wrappers designed manually using the managed C++.
Drawings.NET is a set of wrappers, generated by means of a swig tool - and is a set of C# wrappers, that use pInvoke to address C++ code.
Both of these products allow to use ODA SDK functionality with managed languages, like C#, C++/CLI, VB. These products are set of wrappers for C++ native code.
.NET Classic (or Drawings .NET Classic) is an API which is identical to AutoCAD .NET API. If you have .NET plugin for AutoCAD, you'll be able to build it for ODA apps almost without changes. You can consider using .NET Classic if you are familiar with AutoCAD .NET API, or if you have existing .NET plugin for AutoCAD which you want to port to ODA based applications.
The disadvantage is that .NET Classic covers less C++ functionality than Drawings.NET. You can request to implement a .NET class from AutoCAD API if it is not implemented yet, but it requires more time than in Drawings.NET (because in .NET Classic we have to implement each class manually, while in Drawings.NET we use automatic code generation).
Drawings.NET (or .NET Swig, or .NET) provides API identical to ODA C++ interface. It provides same classes with same functions and parameters as ODA C++ SDK. On this project we use automatic code generation, so almost all ODA C++ functionality is available.