You've probably guessed that the reason why I haven't run a game in this setting using the original rules is that I don't care for them very much. Too much crunch for my liking, too much emphasis on laundry lists of spells and equipment. I know, to some people that's what SR is all about. Not me.
SAVAGE WORLDS worked okay for a while, but my players found it a bit too swingy with the exploding rolls. The characters needed to be a little higher powered to fit the game fiction I'd set up in Disavowed, and I wasn't hugely interested in simply promoting them a Rank or two to compensate -- that would have dropped a whole lot more crunchy bits on the players. FATE provided a better solution, allowing us to concentrate on the game fiction we were interested in, and giving us tools that were better focused on developing unique character than on combat (which didn't come up that often).
FATE provides a robust set of tools that make the SR GM's job a lot easier, Aspects being the most powerful. They can cover a lot of ground on the character creation side, letting players develop unique concepts and tap into favourite archetypes without a lot of heavy lifting. Whether you're a Former Company Man or an Elven Gunslinger Adept, a quick High Concept Aspect pretty much has you covered. When I was developing a set of pre-gen Archetypes so the players could dive into my most recent game quickly, I also made one of the three provided Aspects (out of five) a Tribe Aspect -- basically, an Aspect that connected the character to a location or social group in the city. I gave these evocative names but didn't define what they actually were, so the players could add a little flavour and a piece of worldbuilding to the setting quickly. My Former Company Woman had a Tribe Aspect connecting her to "The Orphanage", which she decided was a strip club. Eww!
On the GM side, Aspects are an awesome tool. If there is something I want to give some weight in a scene, like lighting, weather, or high security, giving it an Aspect makes that important. Better yet, I can defer creating Aspects to the players, outsourcing the hard work on building up a mission and structuring their mission prep at the same time. I simply go around the table once or twice, asking them what kinds of things they're doing to prepare, and let them Create Advantages that represent Gear, Vehicles, information they've gathered about the target, or plans of action once they kick in the door. Boom, you're good to go.
I also gave my Archetypes a full rack of four Stunts, to represent things the characters were good at, pieces of equipment or cyberware, and connections. Every character got a special Contacts Stunt with three NPC types who they could contact for a useful piece of information, once per session. (After that, it cost them a Fate point.) Paired with the Tribe Aspect, this helps create the "underworld networking" element of the SR experience, where who you know and who you owe is often as important as how well you can shoot that smartgun. Also, note that I was working from an assumption that the characters all had 4 Stunts, reducing their Refresh to 2 -- making Compels more important.
I was working from the FATE ACCELERATED iteration of the rules this time, which is one of my favourites (although my recent work with ATOMIC ROBO makes it almost a dead heat). That means that instead of having specific Skills for the characters, I had broad Approaches that cover a whole range of actions. To give these some setting-specific flavour, I re-skinned them as Bold, Cunning, Dark, Hard, Smooth, and Quick. I liked the feel these gave the characters, although they didn't work as elegantly when I was expressing different Stunts (leading to awkward phrasing like "make a Hard Attack"). Briefly, Bold = Flashy, Cunning = Clever, Dark = Sneaky, Hard = Forceful, Smooth = Careful, and Quick is, well, Quick. In play, these worked pretty well, although players would sometimes be confused over which one did what. (If I were to run SR again using FATE, it would probably be with a version that includes specific Skills to remedy this.)
Something I've seen other long-time FATE GMs talk about, even the guys at Evil Hat who designed the system, is how Mooks tend to be boring in play. This is absolutely true, and insignificant encounters are much better handled using a simple Overcome roll, in my opinion. If I want an encounter to have some weight, I stat it up using the "Fate Fractal" / Bronze rule, making it look like a character. So I'd have a Stress track, Consequences, and Stunts in addition to some Approaches for my encounters. This extends the encounter a little but adds some interest - the goons still go down like tenpins, but at least they have a few interesting tricks up their sleeve making them worth the PCs' time and energy.
I didn't spend a lot of my time making specific adaptations of material from the SR canon, because frankly neither myself nor my players were interested in that. We were interested in using the setting as a backdrop for a crime drama full of heartache and hard decisions. FATE worked great for that. Something I've been doing in my other recent FATE game, Not Fade Away, is writing out 3x5 cards for each of the players listing their Aspects. In a dramatic scene, the players exchange them, giving their scene partner a concrete way of pushing on them -- by Compelling Aspects. This works extremely well, and I'd do it for all future FATE games I run.
The only thing that I was a little unsatisfied with, at the end of the game, is that I felt it was often hard to push the characters hard in conflicts and give them a few Consequences for their trouble. One of the characters in particular had a combination of bonuses and good luck that made her avoid so much as a single Consequence the whole game, while another had really gotten some serious lumps. I've been toying with the idea of eliminating Stress tracks in Fate for a while, replacing them and the Consequences with a series of defined Conditions (as found in the FATE TOOLKIT) or else something that looks more like Fred Hicks' recent hack of Star Wars, as seen on his G+ page. It depends how much conflict you want your characters to get into, I suppose, and how you want the game to feel. Standard FATE characters feel pretty capable and heroic, so adjusting this down is going to increase the grittiness of the game overall.
Flavour is the best reason to use a FATE hack of SR over the native rules, in my opinion. The system is quite flexible and hackable, allowing you to put your own stamp on it and make the game work the way you hope it will. Whether it's a crime drama you've got in mind or an over-the-top anime-inspired action comedy (my idea for a new SR game), FATE has you covered.