In order to enjoy these two films, you need to accept the fact they're not even in the same style as the original. The original Psycho is an off-its-rocker horror film made by the premier suspense director of cinema history. The sequels are horror-inflected melodramas, more in the vein of such post-Psycho gaslighting films as Scream of Fear and Whatever Happened to Baby Jane. As such, they resemble Hitchcock's original less than the current, addictive, and feminist Bates Motel.
Both the 1980s films turn on Norman's acquaintance with an odd, slight, half-pretty young woman, a go-between for the audience. In Psycho II, this is Meg Tilly, whose character may not be the rootless outcast she appears to be. In Psycho III, Diana Scarwid is a failed nun. You know this right away, as the film opens with her wailing, "There. Is. No. God!!!" (By that time, Tilly had also played a troubled nun, in Agnes of God. Hollywood values!)
Psycho II is the work of two non-prolific but estimable horror talents, director Richard Franklin (the enduring Road Games) and writer Tom Holland. Holland is a seminal figure in 1980s horror, writer of the original Child's Play, and writer/director of Fright Night. Psycho III uses different filmmakers (Anthony Perkins directed) but makes for a matched pair, with both films competently paying homage to Hitchcock's signature camera work and set pieces.
In 1983, the filmmakers lacked the nerve to abandon the Norman-and-his-mother relationship that shocked the world in 1960, which necessitates character(s) pretending to be Norman's mother (or is it a hoax?). This plotting sets up a franchise, but also lets some of the air out of a claustrophobic universe, making it a typical goth-soaper in which most of the characters are half-nuts. Whereas the original seemed set in the Southwest, the sequel is vaguely Southern.
Psycho II is well-directed, but plays out like a good TV-movie, never as good as you want it to be. (Ironically, Hitchcock shot the original with the crew from his TV show, but at the time, the TV aesthetic meant down-and-dirty and under-the radar. It also meant black-and-white, an ascetic denial suiting the horror genre.) The plot of Psycho II is pleasantly twisting, and gratifyingly sicko, and the ending sent 'em home grinning: that-Norman-he's-incorrigible!
If II is more of a crowd-pleasing roller-coaster, III is small and gritty, but also meaner, involving the deaths of innocents. As before, the script lays on the coincidences, as Norman Bates becomes involved with another fleeing blonde with the initials M.C.
If you haven't seen the movies lately, it's probably impossible to keep their plots separate: which one has Jeff Fahey, and which Dennis Franz? In which does Norman flash back to poisoning his mother? Which murders are actually committed by Norman? This isn't meant as a criticism, in fact the conflation implicitly reflects Norman's mental state.
These are potboilers, sure, but a slash above most horror sequels; they also have compassion for the mentally ill without soft-soaping the symptoms and possible dangers. In retrospect, Psycho was a coming-out film for Tony Perkins, not so much for sexuality but for its implications about his mental health. Perkins already had a successful film career, but had the guts to play possibly the most disturbing character in film history, in a film that many in Hollywood expected to be a disaster (see Hitchcock with Anthony Hopkins). He was typecast, but that couldn't have been too much of a surprise (I haven't yet read his autobiography), and Perkins made a nice career of sketchy loners: The Trial, Pretty Poison, Crimes of Passion.
In the 1980s films, Norman is a serial killer who's spent 20 years in prison; he's also a closeted cross-dresser and voyeur, an isolate, and a socially awkward oddball. Still, his basic goodness shines through. He's trying to cope with mental illness, and with the inevitable mocking and scapegoating that go with it.
Psycho II and III don't try to match one of the best films ever made; like Norman, they have some humility. Norman likes to say "we all go a little mad sometimes." In the original, he's self-justifying, but in these two sequels the world has caught up to Norman: motels have become notorious, and many people are anxious to harass or exploit a troubled ex-con. The original Psycho is famous for making viewers identify with a psychopath, but by 1983, Norman had us with the opening credits. In the current TV series Bates Motel, both young Norman and his smother (a remarkable performance by Vera Farmiga) are sympathetic heroes, despite their massive issues and homicidal ways.
** major spoilers, remainder of post **
In Psycho II, someone keeps calling Norman claiming to be his mother (who, presumably, is still dead). The viewer doesn't know if this is part of the gaslight hoax engineered by Lila Loomis (Vera Miles, returning from the original film), one of Norman's hallucinations, or -- ? Finally Mrs. Spool, a background character, turns out to be the lucky lady: the tag of the film has her visiting Norman at night and revealing she's his mother, although she didn't raise him. Well past his wit's end, Norman poisons her and, before the poison can fully take affect, brains her with a shovel.
In Psycho III, this plotline is put on hold, as Norman resumes his life's work of killing sexy, available women. At the end, a nosey-feminist-reporter-type reveals to Norman that Mrs. Spool wasn't his mother, although in her madness she may've thought she was. Spool was Norman's aunt, and part of a love triangle with her sister and brother-in-law (Norman's parents). Spool unraveled and killed her lover (Norman's father), then spent years in an asylum. Upon hearing this news, Norman fights off his demons in order to destroy his mummy (the preserved remains of Mrs. Bates, or is it Mrs. Spool? I've lost track.) instead of attacking Venable, the reporter. Since Norman has already killed several others, he's hauled off to the puzzle palace.