What is the difference between Cygwin and MinGW?
As a simplification, it's like this:
Compile something in Cygwin and you are compiling it for Cygwin.
Compile something in MinGW and you are compiling it for Windows.
About Cygwin
Cygwin is a compatibility layer that makes it easy to port simple Unix-based applications to Windows, by emulating many of the basic interfaces that Unix-based operating systems provide, such as pipes, Unix-style file and directory access, and so on as documented by the POSIX standards. If you have existing source code that uses these interfaces, you may be able to compile it for use with Cygwin after making very few or even no changes, greatly simplifying the process of porting simple IO based Unix code for use on Windows.
When you distribute your software, the recipient will need to run it along with the Cygwin run-time environment (provided by the file cygwin1.dll). You may distribute this with your software, but your software will have to comply with its open source license. Even just linking your software with it, but distributing the dll separately, can still impose license restrictions on your code.
About MinGW
MinGW aims to simply be a port of GNU's development tools for Windows. It does not attempt to emulate or provide comprehensive compatibility with Unix, other that to provide a version of the GNU Compiler Collection, GNU Binutils and GNU Debugger that can be used natively in Windows. It also includes header files allowing the use of Windows' native API in your code.
As a result your application needs to specifically be programmed for Windows, using the Windows API, which may mean significant alteration if it was created to rely on being run in a standard Unix environment and use Unix-specific features. By default, code compiled in MinGW's GCC will compile to a native Windows X86 target, including .exe and .dll files, though you could also cross-compile with the right settings, since you are basically using the GNU compiler tools suite.
MinGW is a free and open source alternative to using the Microsoft Visual C++ compiler and its associated linking/make tools on Windows. It may be possible in some cases to use MinGW to compile something that was intended for compiling with Microsoft Visual C++ without too many modifications.
Even though MingW includes some header files and interface code allowing your code to interact with the Windows API, as with the regular standard libraries this doesn't impose licensing restrictions on software you have created.
Other considerations
For any non-trivial software application, such as one that uses a graphical interface, multimedia or accesses devices on the system, you leave the boundary of what Cygwin can do for you and further work will be needed to make your code cross-platform. But, this task can be simplified by using cross-platform toolkits or frameworks that allow coding once and having your code compile successfully for any platform. If you use such a framework from the start, you can not only reduce your headaches when it comes time to port to another platform but you can use the same graphical widgets - windows, menus and controls - across all platforms if you're writing a GUI app, and have them appear native to the user.
For instance, the open source Qt framework is a popular and comprehensive cross-platform development framework, allowing the building of graphical applications that work across operating systems including windows. There are other such frameworks too. In addition to the large frameworks there are thousands of more specialized software libraries in existence which support multiple platforms allowing you to worry less about writing different code for different platforms.
When you are developing cross-platform software from the start, you would not normally have any reason to use Cygwin. When compiled on Windows, you would usually aim to make your code able to be compiled with either MingW or Microsoft Visual C/C++, or both. When compiling on Linux/*nix, you'd most often compile it with the GNU compilers and tools directly.
Cygwin is an attempt to create a complete UNIX/POSIX environment on Windows. To do this it uses various DLLs. While these DLLs are covered by GPLv3+, their license contains an exception that does not force a derived work to be covered by the GPLv3+. MinGW is a C/C++ compiler suite which allows you to create Windows executables without dependency on such DLLs - you only need the normal MSVC runtimes, which are part of any normal Microsoft Windows installation.
You can also get a small UNIX/POSIX like environment, compiled with MinGW called MSYS. It doesn't have anywhere near all the features of Cygwin, but is ideal for programmers wanting to use MinGW.