In a fashion, apt-src acts like an apt-get for source packages rather than binary packages, gathering together several commands involving source package management, whereas apt-get source does only one thing i.e. downloads package source code.

apt-get will be here for many moons to come. In a similar way to apt-src, the aim of apt is to gather together the functionality of apt-get, apt-cache, apt-mark etc. in a more cohesive and user-friendly way. I'm not a great fan of it (to me it's a little bit like the "We have 5 competing standards, let's create an all-encompassing one. Ah, now we have 6 competing standards" type of situation), but hey ho.

So what's the difference between apt-src. and apt-get source --compile?

I still use the separate apt-get/apt-cache, as apt itself is missing functionality.

Both apt-get source --compile and apt-src install --build essentially call dpkg-buildpackage to build the package. As to the differences in the ways that they do so (command-line options, checks etc.), I don't know to be honest, sorry.

One of the annoying things with the apt command is that Mint has provided Mint-specific Python scripts to add functionality to the command, so the functionality that's available on, say, Ubuntu isn't the same as that which available on Mint. That's one of the reasons I'm sticking to apt-get and friends (until they can sort out a proper cross-distro implementation of apt in any case). The apt command just seems to me to confuse issues, especially when providing support.

This command will download the source code packages, and build pakage(s) :