Weird fiction can refer to a couple of different types of literature, depending on who's discussing which books and stories.
"Classic" weird fiction is the type of late 19th Century/early 20th Century speculative fiction written by authors such as H.P Lovecraft, Robert Chambers, William Hope Hodgson, Lord Dunsany, Arthur Machen, M. R. James, and Clark Ashton Smith. These stories and novels focus on the supernatural or paranormal and blend in aspects of science fiction with horror. Pretty much any ghost story from that era would classify as weird fiction.
"New" weird fiction is often classified as slipstream or magic realist fiction: fiction that could pass as a regular literary story except for the inclusion of a few details of the fantastic. Alternately, it's sometimes defined as modern literature inspired by the works of Lovecraft, Chambers, Dunsany etc.; this type of weird fiction is frequently written in the style of the old masters but either puts the story in a modern setting or alters/subverts some of the tropes. Lovecraft, for instance, would never write a story that featured a black female protagonist fighting Cthulhu cultists, but I've seen several fine weird stories that do just that.
Still another definiton of new weird stories is that they're richly-imagined, cross-genre tales that feature powerful, detailed world-building. I love this type of cross-genre literature; some people label it new weird and others label as something else, depending on what flavor it is. That's the thing about really excellent stories and novels: they're never just one thing, are they? New weird mixes science fiction, fantasy, and horror up in a brain explosion of hybrid vigor.
What do I love about new weird stories? They're never the same old thing I read last week. They're transgressive. They're political. They're prickly, edgy, cerebral, difficult. Most of all, though, I love the rich, detailed, fabulous, unabashedly weird world-building that some of the best new weird stories have to offer.
The new weird takes you places you've never been before. They transport you. I want a book that'll be like an overnight road trip to some strange new amazing border city with all my odd Goth friends: the car's crammed full of conversations and arguments as the tires beneath you devour the midnight road, and that rattling noise coming from the trunk is unsettling, and that rest stop is utterly creepy, but you get to the city and there's amazing music and architecture and food and drinks and wait, what kind of mushrooms did you say were in that, again?
New weird stories work precisely because of that loving, painstaking world-building. An author has to sell outlandish characters and story elements to his or her readers, and the single best way to do that is to ground a story in vivid, believable details. Do it right, and you've got a piece that's immersive, compelling, and unforgettable.
Horror and dark fantasy readers just don’t get tired of tentacled monstrosities, madness, esoteric cults, and cosmic doom. And why should they? All that’s a heck of a lot of fun. Or it can be, provided writers are willing to come up with new, smart takes on things. There aren’t any bad tropes in weird fiction; there are just tired, unimaginative treatments of them.