Alright, first up, differences between the Director's Cut and the original game - there's a couple of new small scenes added where you play as Nico, including one at the very beginning, that just serve to give some more backstory on her father and his involvment with the shady doings of the game. There's also some new art and dialogue, but honestly, I'm still not sure what's new and what isn't. Also, I saw it mentioned somewhere that some optional "flavor" dialogue was cut from conversations with various minor NPC characters. It seems you can still talk to most of these characters about all sorts of pointless things, however, so I don't know how deep those cuts are exactly, I just saw a couple of complaints around about that.
Broken Sword is kind of in the Gabriel Knight mold, taking place in the present day without a lot of fantastical elements or danger, mostly focused on investigative work to solve a historical mystery. It doesn't have the dark edge of the GK games though, being more light and comical in tone. Main character George Stobbart is something like a cross between Brandon from Kyrandia and a somewhat smarter version of Fry from Futurama in personality. I think his first entry in the in-game journal sums him up pretty well - "I was relaxing at the cafe and a clown blew it up! A CLOWN! Woah! I'll bring that clown to justice if it's the last thing I do!" George is just an American law student on vacation in Paris running his wack game on the French ladies, when he happens to witness the bombing of a cafe by a clown. The Parisian authorities seem quick to want to drop the matter without much investigation, so George just decides to get into it himself for no particularly good reason. He soon crosses paths with Parisian reporter Nico, who apparently was not a playable character in the original game, but in the Director's Cut you'll start out the game with her, and take control of her for a couple of other short segments along the way. Nico is doing a series of stories on the Costume Killer, a guy who has a fairly self-explanatory M.O., and she believes the cafe bombing is one of his. She's soon told by her editor to drop the story, however, and teams up with George to independently investigate, which naturally leads to a whole bunch of buried Templar secrets and conspiracies and etc.
Broken Sword also doesn't use the "virtual theater" engine of Revolution's previous adventure games (Lure of the Temptress and Beneath a Steel Sky), instead using a more standard Sierra/LucasArts style engine where NPCs mostly sit or stand around in the same locations waiting for you to conclude your business with them. This makes for a much more linear and simple effort than their previous games, with the action taking place in a lot of small self-contained environments. I'm not sure if this was present in the original game, but in the Director's Cut, everything you can interact with on a screen is marked by a flashing hotspot, so it's also very much on the easy side of adventure games (you also can't die or irrevocably mess up a puzzle, with only a couple of exceptions.) The only challenge comes from the occasional logic puzzle tossed in (sliding blocks and word searches, for two examples), or when the game is just being obtuse or illogical, which starts to happen a little more frequently toward the end. Fortunately, in those few moments that can be frustrating for whatever reason, there's also an in-game progressive hint system you can call upon at any time for the puzzle at hand (again, not sure if this was present in the original game.)
New voice acting seems to have been recorded for all the new lines, using the original actors; however, there's often a noticable drop in quality at points, with some dialogue sequences sounding like they're very poorly compressed - I assume those were taken from the original game. It doesn't destroy the atmosphere or anything, but it is noticably shoddy.
Broken Sword is a game that I think is strictly going to be for serious adventure game conossieurs. As mentioned, it's easy, which would normally be inviting to adventure newbs, but in this case it's also paired up with a slow pace, heaps of dialogue and very little action. The appeal of the game is in enjoying the detailed Parisian scenery (and later some other parts of Europe), the well-written characters of George and Nico, the dialogue in their interactions with the often stereotypically lazy and lacksadaisical NPCs, the frequently competent voice acting, and the general atmosphere. The puzzles are just too much of a breeze, however, and the overall mystery honestly isn't all that compelling; the more comical bits are where the game really shines, but fortunately that constitutes the lion's share of the 6 or so hours this one takes to complete.
* Gameplay Video