From a production point of view, it is easier to film these conversations having the actor on set. The way typical "comm screen" conversations are usually filmed, the person on the screen is filmed separately from everyone else, and the two actors conversing never have direct interaction with each other.
That used to be the case. These days, we have high-definition video monitors so that the actors really can communicate live over the screen (well, unless they're on redresses of the same set like Shatner and Montalban were). Heck, they had actors communicating live over TV screens as far back as Space: 1999, Battlestar Galactica, and Buck Rogers in the '70s, and even Doctor Who in the '60s -- it just didn't look as good as it does today. Star Trek didn't do it because its producers didn't think 20th-century cathode ray tubes would be convincing as futuristic communication devices, although a lot of other shows didn't mind. These days, though, HDTV screens have feature-quality resolution and are often used as practical viewscreens in SF productions.
With holographic communicators you get the benefit of not having the two people having conversation cut off from each other, they're on set together and can play off each other to better enhance their performance, which was the main argument which allowed the holographic communicator to be used on DS9 when Sisko was hunting Eddington, Avery Brooks and Kenneth Marshall were able to enhance their performances by playing off each other.
Although Discovery stupidly negates these benefits by having the hologram be shimmery and transparent, meaning a visual effect is used on it and extra money is being spent, so they're not really getting ahead anyway.
It's not stupid. The reason DS9 dropped it so quickly is because it was difficult to convey the difference between a person who was physically there and a holographic image, except by having the "hologram" stay motionless inside the projector frame, which was too static and visually uninteresting.
And they are still getting ahead, because it's not about saving money. It's about allowing the actors to play off each other directly and give a better joint performance, and it's about freeing the directors to use a wider range of camera angles than they'd have available for shooting a viewscreen conversation.