Will updating software with a 'deb' file remove the previous version?

Yes. Installing an application from .deb file using dpkg -i will replace the currently installed version with the one specified in the command. Even if the installed one is newer.

This is if the software can be installed with a single deb file (like the atom text editor).

But if it requires more packages to be pulled to be installed, then you may need to do sudo apt-get -f install after dpkg -i command. However, in this case, if you already have newer versions of those other packages, you may need to force package managers to downgrade these versions. Otherwise, apt-get -f install will end up installing the newer one.