The biggest thing would be immortality. People without souls don't go to Heaven or Hell. Should she die she would cease to exist. A soul in this instance is sort of an insurance policy.
The lack of soul also hits her on an emotional level because of her relationship with her mother Lilith. Because Lilith isn't God, she was unable to create souls. But she decided to create demons anyway which Mazikeen seems to resent. She feels incomplete because her mother basically created her to be a torturer in hell and have no other purpose. Now that Mazikeen is no longer in Hell, she has no direction or reason to exist. She may feel in this sense, that a soul would make her "whole" or "complete".
And finally a current theory, as in this is currently not verified in the show but it might in Season 5 part 2, is that Mazikeen has already developed a soul but is not currently aware of it. Prior to the events of the show and in early seasons, she was pretty "soulless". She didn't care about anybody. She was pretty self-centered with the exception of her undying loyalty to Lucifer. However, now she exhibits all the traits that you described in the OP. Her new found emotions may be the reason she suddenly cares about whether or not she has a soul.
The show has finished and there is a big spoiler in Season 5 Part 2 below.
It has been confirmed that Mazikeen did, in fact, grow a soul (or at least God implied she did, she jumped to that conclusion, and they never said it was wrong.)