And, in no particular order, some cultural aspects of a non-Freudian modern world...
I think psychology, which predated psychoanalysis, would still have gotten around to discovering talk-therapy, albeit with different focuses than Freud. The humanistic Carl Rogers/Bob Newhart type might be the stock image, rather than the half-mad central European with the couch and cigar.
I think the idea of an Oedipal complex would still exist in one form or another, as academics would notice the theme throughout art, literature, religion, etc. It might be something more confined to elite discourse, rather than the subject of popular reference.
People woud still talk about Freudian slips, though obviously under a different name. I can't imagine that Freud was the first person to notice that.
The phrase "anal retentive" would not be in everyday use, though a phobia of defecation(not neccessarily connected to childhood) would still be associated, here and there, with a certain type of personality. "Yeah, my boss, what an uptight WASP. Kinda guy who has to be forced at gunpoint to take a dump."
I think something very similar to Jungianism would still come into its own, though obviously under a different name. With the collapse of religious authority, there would likely be a longing for a religion-infused wordview, operating within at least partially scientific parameters.
I'm tempted to say that the thwarting of Freudianism butterflies away the surrealists, but I'm not so sure. The idea of taking dreams for artistic inspiration goes back before Freud, and would probably have been revived eventually.
But Hollywood output would be radically altered. The absence of Freud butterflies away much of Hitchcock, Kubrick, and Woody Allen, just for starters.