Each game has a different story.
They're all worth playing.
My personal favourite is Dragonfall, but if you're going to play them all I'd recommend playing them in order.

Start with the SNES game.

Play them in order, because the developers learnt as they went. I quite like the aesthetics of the first engine (especially the drone controls, etcetera, around the screen). The first engine is more lenient with combat; being struck without cover is not as fatal as in the latter engines.

Dragonfall is the best game. Hong Kong introduced a third engine. There is a sizable collection of User Generated Content, though because of the triple release (three engines for three games), most mods work on either the first or the third engine (which is notable because of the [i}Harebrained[/i] retcon of matrix combat). There is a fan mod for the first game, built for the third engine, for example: Vox Populi.

I found the third game both easy and filled with frustratingly pointless conversations. Sure, they add some depth, but they add it with characters that only appear once. So there is no loss if you never talk to them. (Certainly the conversations are not worth it.) The bonus levels, accessed only with a PC who has completed the main quest, is worth it (but high-level difficult).

Dragonfall, however, starts with a great set piece and the best music in the series. (There is a mod to replace the horrid HK soundscape.)

All comparisons are odius, it's true, but I would probably rate the Antumbra saga above HK. (And Vox Pop, and a bunch of others, too.)
The first one is for the Dragonfall engine: Antumbra.
The next two expansions are both built for engine three: The Caldicott Caper and CalFree in Chains.

There are some other mods, too (notably for the first engine) which are all of high quality.

edit: hyperlink syntax

I think Dragonfall was originally an add-on for Returns. I was near the end when they released Dragonfall: Director's Cut.
It was made standalone for the Director's Cut and many improvements were made.
The Dragonfall with the Shadowrun Returns backups is the non-director's cut. Many improvements were introduced in the Director's Cut.
Each title has its own story and modifications to the base game
So each standalone game alters the base game differently.

Ja, the Director's Cut allows a player to opt for the evil alignment solution (killing rather than saving).

The UGC is unfortunately fragmented, though, because there are THREE game engines (with three different versions to manage), and each world commanded by these engines is slightly different. The last engine radically changed the matrix, for example (transgressing and turn-based combat, both).

This means there are more mods ("User-Generated Content", or UGC) for the first engine, almost nothing —because the third game was released very soon after— for the second engine and a couple for the third.

I worked my way through all 3 games in order, currently on the 2nd adventure in HK. Thoroughly enjoyed/enjoying all 3 games. I would play them in order because (as mentioned earlier) the game engine evolves a bit each time. The 2nd game engine was my favorite because it had more strategy in the fights (cover and partial-cover are more important), but always remember you can adjust the difficulty of the game at ANY point, including before a fight. The 3rd engine, the only thing I didn't like was the mini-games that the Decker would encounter while in the Matrix...I had very little idea what I was doing, but you can skip past those with a penalty that doesn't ruin the game.

The SNES game was an arcade/console-styled game and not a CRPG by any stretch of the imagination. The SEGA Genesis Shadowrun game was a CRPG (in as much as a real-time game can be) and even used most of the pen and paper Shadowrun RPG rules for characters.