(or "No, I Won't Compare It To 'Savage Sword Of Conan'")
'Mystery Science Theater 3000' (and 'The Film Crew' and 'Cinematic Titanic', its spiritual heirs) has a luxury that a lot of series don't possess; it doesn't really need a storytelling engine at all. The core of the concept is "comedians delivering a humorous commentary track to an existing movie", and Rifftrax shows that deep down, that alone is enough to deliver entertainment value. The extraneous elements--the sketches, the characters, the rationale for making fun of cheesy (and not-so-cheesy) movies--all that is just icing on the cake. This means that, since it doesn't necessarily have to carry the series every time, the writers of the series can and have tinkered with their storytelling engine quite a bit over the years--sometimes out of necessity, sometimes just to improve the comedy.
When it started, MST3K had a fairly involved backstory (much of which never made it to the screen.) Joel Robinson and Clayton Forrester both worked at Gizmonics Institute, a sort of "not quite mad science" university where people greeted each other by showing off their latest inventions. (Yes, that's what the Invention Exchange is supposed to be every week.) Doctor Forrester and his fellow scientist Dr. Laurence "Larry" Erhardt went renegade, hiding in a self-made secret lair called "Deep 13" and deciding to conquer the world by inflicting bad movies on people until they begged for mercy. Forrester chose Joel as his test subject, based primarily on an irrational dislike of him, and shot him into space to begin the experiment. Joel, in turn, built a series of robots out of non-essential bits of the ship to keep him company.
So that's the dynamic at the start of the series. Two mad scientist buddies on Earth, and Joel as the slightly-bemused father figure to a trio of wise-cracking robots (plus Cambot, who never talks, and Magic Voice, who doesn't have a body.) Fairly simple, but you can already see room for improvement. Doctor Erhardt doesn't really have much to distinguish him from Doctor Forrester, and there's a certain "comedy villain" dynamic that they're struggling to establish; the villain has to be evil enough to be credible as a villain, but can't actually succeed because failure is funny.
Luckily, necessity became the mother of invention as J. Elvis Weinstein, the actor playing Erhardt, left after the first season. Frank Conniff replaced him as "TV's Frank", and in so doing provided Doctor Forrester with exactly what every comedy mad scientist needs--a bumbling assistant. This freed Doctor Forrester up to become the cartoonishly evil mad scientist he needed to be for the success of the series, because Frank could foil his schemes through his sheer incompetence in carrying them out. (Why doesn't Doctor Forrester find a better lackey? Just repeat to yourself it's just a show...)
This dynamic continues through Seasons Two, Three, Four, and part of Five, and proves to be an extraordinarily stable generator of comedy sketch ideas. The robots settle into their personalities fairly quickly (Tom as pompous blowhard, Crow as exuberant man-child, Gypsy as the seemingly dim-witted one who actually has all the common sense), and the whole thing runs quite smoothly...
Until Joel Hodgson, series co-creator and the actor playing Joel Robinson, decides to leave the series. Suddenly, they're without a human host, and without a key part of the dynamic--Joel is the principal foil for the Mads, he's essential to the rationale for the series, and he acts as an authority figure to the bots (which paradoxically allows them to act out more, not less--having Joel there to put the brakes on their antics means that they can constantly push those boundaries themselves.) The series obviously needs another host, but the exact mix of elements Joel provides is irreplaceable. So how do you solve this?
Enter Mike Nelson, playing...er, Mike Nelson. Mike trades in Joel's slightly-bemused father figure role for a completely bewildered bachelor uncle, or perhaps older brother...he didn't build the bots, he's not an authority figure--or at least not one they're willing to consistently recognize. His efforts to assert some form of control over the rebellious bots, and his frequent failure (because failure is funny) becomes the new dynamic and source of comedy for the rest of the series. This gives the Mike episodes a slightly "edgier" tone, because Joel isn't there to act as a brake on the darker comedy and Mike can't fill that role, but it's nothing you can't see shades of when you look back at the older material.
Meanwhile, back on Earth, Frank Conniff leaves the series, to be replaced by Mary Jo Pehl as Doctor Forrester's mother, Pearl. Pearl doesn't work very well as a second banana, for the same reasons that made her so entertaining as an occasional guest star; she's the next place up on the food chain, the figure that makes Doctor Forrester feel just as helpless and incompetent as he makes Frank feel. With her in place, suddenly he looks like the bumbling sidekick, and if there's one thing a series can't have, it's confusion as to who's the lead villain and who's the lackey.
But after six episodes (Season Seven was quite short), Trace Beaulieu bows out and Pearl becomes the lead villain. They introduce not one, but two sidekicks for her--Professor Bobo, who begins as an intelligent ape scientist but who loses about fifty IQ points an episode (because bumbling sidekicks are funnier, see above) and Observer, who actually fulfills the role of "smart, competent villain" but clearly takes his orders from Pearl (this is known, in some circles, as the "Jeeves and Wooster" comedy dynamic.) It's this final model that runs through the last three seasons of the series.
All these are mainly just variations on a theme, but it's interesting to note which variations find their rhythm and which get tinkered with over the course of the series. As the subsequent direct-to-DVD series (again, 'The Film Crew' and 'Cinematic Titanic', both very worthy successors to the original) show, vast chunks of the concept can be changed, added, or jettisoned while still retaining a good comedy dynamic ('The Film Crew' takes a mix of bored office-workers in a sinecure job and friends doing some male-bonding over a bad movie, while 'Cinematic Titanic' has vaguely conspiratorial overtones as Joel and friends reunite to save bad movies for posterity.) The core of the series, when done well, is always so vital and entertaining that the fans can enjoy a little experimentation.