Cosmic or Lovecraftian horror is one of scale and weight. The motives of the entities we face are so far reaching and complex we can't comprehend what the end game of whatever we're fighting is.
Its common to have themes of madness and fear of the unknown within its media. Often the entity is inter-dimensional and doesn't follow our logical biology instead having something weird and doesn't make sense. The Mind Flayer from Stranger Things for example is a shadow. its something we can't pin down if its even "real" like can you touch it?
Brethren Moons from Dead Space
SPOILERS FOR SEASON 3 OF STRANGER THINGS YOU HAVE BEEN WARNED
The mind flayer hasn't spoken, we don't know what its intentions are, who it is , does it have a name? beside the one we gave it off a D&D manual? Does this thing perceive things the way we do? All we know its its wants us, and that is very powerful. We're also nearly helpless to a being beyond our comprehension and ability to measure.It most likely isn't going to do kind things to us that is Cosmic Horror
So how might this dread on anxiety Cosmic Horror manages to do be any different from other horror like media?
To make something horror you must make someone be scared duh. Tension ,mystery and fear of the unknown are common themes inside of horror. Fear of the dark and unknown is nothing new. Now what if that darkness in your room at night could plot against you? the very void plots to consume you. Forces outside the control the the human race have sought our demise. Inter dimensional horrors whose bodies, faces (or lack of) and behaviors lack sense but to fulfill one purpose, and it doesn't concern you.
Vitims of the mighty Cthulhu, cruel AM or cunning Mind Flayer aren't killed or eaten by a werewolf or serial killer. They are instead stripped of their sanity and identity to fulfill the purposes of what ever entity the character has encountered. you're not a consequence to these forces of power, but a resource to be harvested at their whim.