WotC's handling of the Far Realms has IMO suffered from severe overexposure over the past several years (I adore Cordell's late 2e work, and find it seriously inspiring, but I swear he hasn't written about anything without tentacles for years now).
That said, the material all too often seems to fail at being any different from the Abyss with slime and tentacles and Lovecraftian pastiche. You need to differentiate from what the Far Realms represents versus what the Abyss represents.
I think the real problem here is that the Far Realm was tacked on after the fact. 3E had the "aberration" creature type, but it wasn't well defined and basically meant "weird stuff with tentacles or excessive numbers of eyes." It did, however, include several iconic D&D monsters such as the mind flayer, beholder, carrion crawler, and aboleth.
So the Far Realm was introduced as a kind of unifying back story for the aberrations, to provide a concrete meaning for the term. Unfortunately, many of those iconic monsters already had a well-established position in the cosmology. Mind flayers especially have an extensively developed culture and play a big role in the politics of the D&D world; they're the chief rivals of the drow for control of the Underdark, as well as being the ancestral nemesis of both the githyanki and the githzerai.
What that meant was that the Far Realm's ability to be unique and distinctive was limited by all this established aberration-lore. Far Realm creatures had to be able to set up comprehensible and functional civilizations in our reality, or mind flayers wouldn't fit. They couldn't be distinguished by their psionic powers, or beholders and carrion crawlers wouldn't fit. The Far Realm had to include some analog to water, or aboleths would make no sense. Et cetera, et cetera. Pretty soon all you could say about the Far Realm to distinguish it from our reality was, "It's really... uh... WEIRD."
This is one of the reasons I prefer to associate aberrations with the Underdark. It's a much better fit IMO, since the one unifying trait of the iconic aberrations is that they were invented back in the day for use in dungeon crawls, which means they're all subterranean. And the hallucinatory, Alice-in-Evil-Wonderland weirdness of the Underdark perfectly matches the nightmarish weirdness of aberrations. Moreover, everything else that lives in the Underdark can be explained as offshoots of surface-dwelling species (e.g., drow) or extraplanar beings (e.g., xorn).