Most stories have a beginning, middle, and end. Even a show about nothing has something zany and inane happen to our characters every episode, while getting some amount of resolution by the end. However, the same isn't necessarily true for a series as a whole. Some series are so homogeneous in plot you could air a season 1 and 5 episode side by side without telling the difference. Others have such intricate plots, you can tell which quarter of which season you're watching just by looking at the subtle nuances of the main couple's relationship.
To quantify this, the Sliding Scale of Season Transition Fluidity (Season Fluidity for short) puts episodic series on one end, and series with self contained seasons on the other. For example, Gilligan's Island is unchanging from season to season. Toward the opposite extreme, seasons in Sailor Moon and Blackadder are basically separate shows with an identical cast (and some shows don't even have that commonality between seasons — see Skins, below). In the middle, a show like Stargate SG-1 has no distinct seasons, but is threaded together by multiple subplots while staying episodic.
Put another way, you can watch any episode of Gilligan's Island and be equally entertained, without worrying that you've missed important plot points (it's not like they'll ever get off the island or something). While that's also mostly true with Stargate SG-1, seeing more episodes in order lets you see character development over time and several subplots rise and get resolved, letting you get more enjoyment over time. A slightly-less fluid series, like Buffy the Vampire Slayer or Mad Men, needs to be seen with all the seasons in order if one wants to fully "get" it. Further toward the abrupt end, you can start with the beginning of any particular season of Sailor Moon, but you will be left scratching your head if you start in the middle of a season.
The far abrupt end would feature shows that do a complete series reboot with each season, in both plot and cast, to the point where they really are completely different shows joined under the same name. An example of that would be Skins, which starts with a completely new group of characters (save an occasional Ascended Extra or two) and storylines every two seasons, as the old characters graduate from Roundview College and leave Bristol. Other than the setting and the general focus on sex, drugs and parties, each new "generation" (as these two-season sets are known) is completely distinct from the others, and many Skins fans see them as entirely separate works. Lots of fans of the show start with a later generation and then watch the older ones, and are no worse for the wear.
Note: For simplicity's sake, this scale is excluding Genre Anthology shows, sketch shows, and other works where there is no status quo to either follow or violate, or semi-consistent set of characters/themes. If these works fit on the scale at all, they would mostly fall on the extreme Fluid end.
Sliding Scale of Season Transition Fluidity
|Fluid||Status Quo Is God on the series level, even possibly including Negative Continuity||Most Golden Age and Dark Age Western Animation plus more recent works that follow those formats (e.g. Animaniacs, Tiny Toon Adventures), most traditional SitComs, South Park in its early seasons|
|All stand-alone episodes, no arcs, but with some degree of continuity. (e.g. dead characters stay dead)||Star Trek: The Original Series, early seasons of Star Trek: The Next Generation, and most other shows that follow the Adventure Towns or Monster of the Week format. Most adult animated comedies, including Family Guy, The Simpsons, and South Park in its later seasons. Most modern Sit Coms. PAW Patrol in its early seasons.|
|Usually stand-alone, but occasionally has arcs||Law & Order and its various spin-offs, The Vicar of Dibley and other standard Brit Coms, most Millennium Age Western Animation, Cowboy Bebop, the Classic series of Doctor Who, PAW Patrol in its later seasons. Typical abruptness limit for Western Animation and SitComs.|
|Series-long Myth Arc||Babylon 5, the Revival series of Doctor Who, most anime and Noughties Drama Series|
|Dammed||Multiple smaller arcs not directly tied to seasons||Star Trek: The Next Generation (later seasons), Glee, Xena: Warrior Princess, Haruhi Suzumiya, Stargate SG-1 (except for seasons 8 & 9), most Soap Operas|
|Self-contained season-arcs, with some overarching plots||24, Mad Men, Game of Thrones, Star Trek: Deep Space Nine, Damages, Buffy and Angel, Skins (seasons with the same cast), most Teen Dramas and Prime Time Soaps|
|Highly-distinct seasons with Arc Welding||Sailor Moon, Pokémon, Power Rangers, Digimon (first two seasons), Boy Meets World|
|Highly-distinct seasons, purely self-contained||Lexx, Blackadder|
|Abrupt||Complete series reboot each season||American Horror Story, Skins (every two seasons), Digimon (third season onward), Super Sentai|