They're completely different animals. An exe file is a window executable format, ie a single compiled windows program. The Linux equivalent is an Elf file but usually has no extension. The mac version is a mach file.

Apk is a package. It contains both executables and resources like images, data files, etc. It is designed to be installed on android where the contents are installed in the correct places. The windows equivalent is usually an MSI file for the windows unstable but there are other packaging systems. On Linux, the common variants are rpm for RedHat and deb for Debian and descendants like Ubuntu but there are many many variants even down to simple zip or tar files.

I don't know the specifics of Apk files but because android is a descendent of java I suspect they are very similar to java jar files which in turn are closely related to zip files

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