I have the old versions, and played the heck out of them.
I don't have version 4. If I buy it, what can I expect?
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We have been playing a Shadowrun 4th Edition game for a very long time now. I find the rules system to be much easier to understand/operate, and the dice rolling is pretty straightforward. I have been playing Shadowrun since 2nd Edition, and personally, I think 4th is the best yet.
Setting wise, the biggest new addition (as Numenetics stated) is the wireless Matrix. Also, now that machines are talking with WiFi, so are Meta-Humans - The Technomancer, a "hacker" that uses a Magic like stat called Resonance to "Thread" bonuses to programs, and summon "Sprites" to do their bidding is now a character option.
There are a lot of great optional rules, like taking "automatic successes" (your dice pool / 4 = number of hits) which allows you to quite easily run a diceless forum game alongside your sit down game. We have taken full advantage of that system in our very long campaign...
The main system difference is that the "Target number" for rolling is ALWAYS 5... I.E, 5 and 6 are the only #'s that ever "hit". 6's do not "explode" like in previous versions unless you use your "Edge" stat. Also, instead of increasing or decreasing target numbers, things increase and decrease Threshold/Pool size.
I would be happy to try to address more specific inquiries into the 4th edition as well, if there is something you were interested in knowing that I didn't mention...
I don't have a lot (i.e., any) experience with previous editions, although I did play the Sega game. :-) That said, some of my fellow players when I played 4th pointed out the big differences that the change to wireless technology made to gameplay. Most deckers no longer jack in and go comatose. They can continue to interact with the rest of the party while hacking. From what I understand, it used to be that when the decker jacked in, everyone pretty much checked out and went to get pizza or whatever.
I've played a little of 3rd edition and a little of 4th edition. I preferred 4th edition for a few reasons:
There are probably other changes that someone more familiar with the game would notice, but those were the big ones I saw.
Shadowrun 4th edition has switched from its original mechanics to a (new) World of Darkness style mechanic. These mechanics are substantially less intricate mathematically, and as such are easier to resolve fairly quickly during gameplay, but are considerably less flexible.
For example, in SR3 and before, tactics of how to spend dice pools were important. In SR4, there aren't any variable dice pools. Result: faster gameplay, but a system which is more stat-dependent and less dependent on player cleverness.
As another example, SR3 works pretty well with amazingly powerful characters, and can do a halfway decent job with very weak ones, but the price you pay for that is having to keep track of a wide range of different target numbers (to differentiate between easy and difficult tasks) as well as the number of successes (to differentiate between barely succeeding and succeeding massively). In SR4, the target numbers are all 5--so gameplay is faster (one less thing to keep track of), but the mechanics no longer really support weak characters doing pretty easy things or extra-powerful characters doing fiendishly difficult things.