Midi files are pretty much just sequences of notes, and contain minimal instrument data - specifically, they have instructions related to what instrument should be used, but don't actually contain the instruments - those have to be provided elsewhere. As such, they sound incredibly low quality on most consumer setups, particularly as Windows' built-in midi instruments are not very good (I believe they're rather lossy compressions of samples from the Roland SC-55, but I'm not 100% certain; they do sound quite similar, however, but the SC-55 actually sounds somewhat passable). They can be used as inputs for equipment/software that makes actually good sounds, which is how they've managed to remain relevant, although my understanding is that generally you wouldn't want to release the raw MIDI itself; rather, you want to release the output from your fancy setup that, you know, actually sounds good.
.mod and related formats such as .it and .xm have sequence data, but they also have associated samples/instruments. This makes them larger than .midi files and smaller than a recording. At one point, these were great - you had more control than with .midi, but you didn't have to use the sheer amount of storage space a recording would require. The fact that you're just using a few samples and playing them back at different rates, however, imposes a limitation on the quality of the output. It's generally better than what you get out of MIDI on a typical consumer setup, but now that distributing recordings generally isn't an issue, so now you can do all your recording on some fancy high quality set-up and then just distribute the recording. And if you just want to use a sequence of note data to control your fancy piece of hardware/software, MIDI is significantly more widely supported. There ARE some newer formats (e.g. .mptm) that are similar to .mod but have stuff like VST support, but the VST plugins complicate distributing them in their original format, so it's often more practical to just distribute a recording.
As such, the main niche for the .mod format is for making stuff that sounds kind of like an old Amiga game. There's certainly more you can do with it than that, but at a certain point you leave what's practical and instead are clearly pushing the limits of what the formats are capable of, which can be an art in its own right, but if you're just trying to make quality music it's usually more practical to use more up-to-date methods.
It shouldn't be surprising that it's possible to find these files for Amiga games, however, because mod-like formats were extremely common for Amiga games. They were a good compromise between the small size of midi sequences and the quality you can get from a recording, and the Amiga's sound hardware was particularly well-suited to it.
