This info seems to be fully applicable:
https://ostechnix.com/how-to-boot-into-rescue-mode-or-emergency-mode-in-ubuntu-18-04/
Based on the text, I take emergency mode to be "a root shell", period; nothing else initialized.
Rescue mode on the other hand has tools to help you get started handling the computer in a reasonable manner - still as root though.
I generally use method 2 (i.e. via the grub menus), actually doing a fsck and then network activation via the final level menu, before dropping to a root shell.
By then, at that root prompt, you have "normal" possibilities to run e.g. apt and all that: e.g. having a display driver problem; remove current driver, find anew one and install it (or reinstall the same).