SDK only includes the necessary building blocks for developing applications. This includes frameworks, libraries, header files, whatever as well as compilers, debuggers, and various other tools, such as profilers, etc.
IDE simply makes access to those more user-friendly (or integrated, hence the name), if you will. However, Eclipse for example comes with its own compiler as well.
In any case, an IDE allows you to develop applications from a single environment, be that Emacs, Visual Studio or Eclipse. If you only have an SDK you'd write programs with a text editor, compile them with the compiler (instead of hitting F\d+), debug with the debugger which often has an awkward text interface, &c.