I read somewhere that Neo Geo Home Version of MetalSlug had special side missions and challenges or something like that.
you read wrong.
For AES/MVS the roms are (for the most part, there are exceptions) the same.
There are different code paths depending on the bios being used, which in the case of the home bios will mean you're presented with an options menu, where you can select the difficulty, and given a limited number of credits to complete the game. If the arcade bios is used then all the options (difficulty etc.) are selected by the arcade op, and the number of credits you get depends on how many coins you insert.
The physical carts are incompatible, but you can buy adapters to run either on either, because the actual content is the same, and will operate in arcade / home mode depending if you're running on an aes/mvs motherboard.