Can't comment on JimB's answer, so tackling it here, in particular the point about Windows integration. Cygwin being based on Win32 instead of running in its own subsystem, while slower, provides much greater Windows integration. It even allows Windows and UNIX APIs to be used in the same program, which enables things like the Cygwin X server or the mintty terminal.
Cygwin's 'ps' does list Windows processes if you give it the -W option, and 'kill -f' will kill them. Windows programs can be invoked from within Cygwin and plugged together with Cygwin programs using all the usual mechanisms such as pipes. (I don't know whether that's possible in Interix.)
Windows-style paths, both with forward and backward slashes, are supported. Cygwin 1.7 made UTF-8 the default character set, and Windows' UTF-16 filenames are automatically translated, thus filenames in any language show up correctly in Cygwin. I don't know what Interix does here, but couldn't find any evidence of it supporting Unicode.
Other integration features include the 'cygstart' utility for opening a file as if you double-clicked it in Explorer, and the /dev/clipboard device for accessing the Windows clipboard.