You can use Media Player Classic's DSM converter to demux the subtitle file.

If you have Media Player Classic installed, the option is available through File > Utils > DSM Converter. Just add the video file, right click on the subtitle stream and choose demux. There're other more speciallized muxers out there so you can reinsert the subtitles back, but I afraid I'm not an expert on that department. You can get MPC-HC in the link below:

http://mpc-hc.sourceforge.net/

You can't get the subtitles directly from AVI or MP4, they're already hardsubed into the video itself. You'll need OCR softwares to capture the subs. Again, I don't have any knowledge about any of these softwares so you have to search yourself for it.

It's impossible for the subtitle to disappear in a hardsubed video as the subtitle is a part of the video stream itself, so it must be softsub (subtitle file embeded in the .mkv file). It seams like both of the players uses different filters to render the subtitles.

Anime fansubs are usually optimised for the CCCP which uses Media Player Classic as the default player, playback problems might occur if other players/ codec pack are used.

Aegisub