We have both, a late mk1 (2008), and a mk2 car.
The mk2 is a nicer car, and looks more modern, particularly the interior - but on closer inspection, the materials used on the mk2 are inferior. Some of the plastics used are quite thin, and it just doesn't feel as robust as the mk1. Silly things like the extending roller boot cover on the mk1, which was replaced with a flimsy folding flat thing, which feels really cheap and nasty. The seats in our 170,000 mile mk1 looks like new, I don't think the mk2 will make anywhere near that mileage without showing some wear, although they are a bit more supportive and a slightly better shape. A few other nice touches disappeared on the mk2, like the levers on the top of the front seats to move them forward from the back.
That said, the mk2 is a nicer and more relaxed car to drive. Although it is in many ways very similar, it is a bit quieter with better isolation from road noise, and the improved steering makes a big difference. The steering on the mk1 has always been a problem, assistance is poor, and it feels reluctant to centre at low speeds. The mk2 feels much more 'normal' and modern, much sharper, with better assistance and less turns lock to lock. The mk1 is fine when you get used to it, but the feel of the mk2 is much better.
Although the mk2 is slightly bigger, it doesn't feel any more spacious inside, and the boot is a slightly less useful shape - I think it's a bit deeper, but not quite as wide. If you go for a mk1, look for a SE which has better spec - alloy wheels, 4 elec windows, rear speakers, and most importantly, aircon.
The mechanical build quality is about the same on the two cars, and most of the mechanicals are very similar. The problems some people have with notchy steering on the mk1 seem to have been fixed with the mk2 set up. The ABS modulator, another common mk1 failure which often resulted in an expensive replacement, is now fitted with a much bigger motor unit, so hopefully that should last. The 2008 mk1 cars were built in China, the mk2 cars here in Swindon, if that makes any difference to you.
The 1.4 i-DSI engine in the mk1 feels more tractable at low revs than the 1.4 16V V-TEC in the mk2, but runs out of puff when extended. The 16V V-TEC engine in the mk2 certainly doesn't feel as nippy around town, but comes into it's own when you can get the revs up. MPG is about the same, maybe a fraction better on the mk2, and tax on our mk2 is a band lower for the same engine size, £20 cheaper. Insurance is about the same.
As I said, the two cars are very similar. I asked the other half, who drives the Jazz full time, which she preferred - she said she would go for whichever was in the best condition, so really, she likes them both. Personally, I think there are a few points which swing it for the mk2 - the more modern looking interior, exterior, better options, and the improved steering.
Sorry it's a bit long winded, but I think that's about it. Hope it helps.