Mono is the open-source version of Microsoft’s .NET Framework. It implements most of the backend framework features, but unfortunately, falls flat on its pretty little face when attempting to display the user interface – which is what desktop apps are all about.
Wine on the other-hand, is a Linux port of (major parts of) Microsoft’s Win32 library – the core dependencies of the Windows development libraries, and more importantly, the win32 interface elements. With Wine, you can run many traditional C++ win32 executables on Linux, with certain limitations.
Mono’s biggest stumbling block is the GUI and .NET programs that use P/Invoke to call native non-managed win32 dlls – Mono is a pure .NET environment, and can’t handle them. But from the description above, that’s exactly what WINE excel at… So can’t we use WINE + Mono to make just about any .NET program run on Linux fresh out of the .NET compiler?
Unfortunately, the answer is no. Back when the Mono project was first starting out, the Mono development team considered using WINE to implement the System.Windows.Forms namespace of the .NET Framework (which is practically 100% native C++ unmanaged win32 code in .NET wrappers). But they made the right choice in deciding to not take the easy way and go that route, leaving the integrity of the Mono project intact and focusing on true cross-platform user interface libraries instead (the GTK# is now the UI Library of choice for cross-platform .NET applications).