We have two MK2s in the family which I drive frequently and my wife has an MK3 sport so can give you my thoughts.
Starting with practicality, I would say there is no meaningful difference. The MK2 seems to have a bigger boot. We went from an MK1 to our MK3 and noticed a slightly smaller boot in the MK3, although the rear passenger legroom was better. The roof level seems lower in the MK3, certainly if you're tall and sitting in the rear side seats, the curving roof might just brush your head. So I would say headroom in MK3 is not quite as good but it isn't bad.
The ride quality is very similar too. I would say MK3 is probably a touch more refined but I've noticed that it seems a little less stable in crosswinds on the motorway compared to our MK1 or the MK2. But as the Jazz is a small light car, it is always going to be unstable in crosswinds. The seats in the MK3 seem firmer, definitely firmer than the MK1 and probably a little firmer than the MK2 which for me is not as comfortable on longer journeys.
One thing I don't like about the MK3 is the front headrests. They are awful. They tilt forward too much and sometimes if accidentally pushed forward they lock in place pushing your head forward. They are also far too big and block the view for rear passengers.
We have the 1.5 engine in our sport which has a decent amount of torque and makes the drive much more pleasant but is slightly thirstier - not that we noticed the difference but on paper, it is not as frugal. I haven't driven the 1.3 MK3 so cannot comment on how it compares with the MK2 equivalent. The 1.4 MK2 isn't bad, a marked improvement in driveability over the 1.4 MK1 which had very poor torque, the 1.5 MK3 with a slightly bigger displacement is better still.
The steering is a big difference between the MK3 and MK2. The MK2 has the worst steering feel of all the Jazz's I've driven. I think the MK1 probably has the best feel and sense of control followed by the MK3.
We've had no major issues with our MK3 and there has been no issues at all with the MK2s in our family. Really reliable cars. The only minor issues we have had have been a squeaky clutch pedal (which was easily fixed) and now a squealing clutch which doesn't affect the driving at all.
In terms of positives, the MK3 looks better in my opinion, it's a design that will likely stand the test of time and will take a long time before it looks dated. The interior is much cleaner and modern than the MK2 and is generally a nicer place to sit. The MK2 interior does look a little old now albeit perfectly useable and practical. There are more "toys" in the MK3 if that's something you're looking for. We don't really use half the features that come with the car. Cruise control is about the only thing we use, great for reduced-speed motorways - but admittedly it is not that easy/intuitive to use compared to other systems.