lots of differences.. let's see, where to start........
one of the main differences is that the "points based" character creation was introduced in SR4. Prior to that, the game used the Priority system.
the very way in which 'hits' are calculated when rolling dice is different. In SR3, a task would be given a difficulty number that would range anywhere from, say, 3 to 20. You'd roll your dice, and if you got ANY dice that were equal to or greater than the difficulty threshold you were successful. If the threshold was over 6, you would re-roll all the dice that had come up as a 6 and add the new roll to the previous. Therefore, meeting a threshold would get progressively harder as the number went higher, to being nearly impossible at the upper end. In SR4, difficulties scale more linearly with the number of hits needed for a success versus the number of dice in your pool.
The way in which skills interact with attributes is different in SR3 as well. Attributes don't directly affect skill ratings as they do in SR4, but instead go into a 'dice pool' that can be added to skill rolls. This means that if you're in melee combat, you can attribute more dice to offense and less to defense, or vice versa. Made combat a bit more strategic, IMO.
Mages used to be in two distinct camps, Shamanistic and Hedonistic. Now they're lumped into the same set of rules, and it's up to the player to role-play the difference.
The real difference in terms of playability is that I found SR3 to be 'grittier' than SR4. You can die a million ways in either system, but it seemed to be a LOT easier to get dead in SR3.
Personally, I actually liked the feel of SR3 better than SR4, but they streamlined a LOT of stuff that was really clunky in SR3 and therefore most people have accepted it as the norm. Hacking was sooooo tedious in SR3, and the rules for rigger vehicles were largely unintelligible and often contradictory.