Hi guys, i'm leaning towards the Spector at the mo. It's £40 cheaper, has a better, more adjustable bridge, Active electronics, much more varied control over the tone, etc
The only thing I can think of to choose the Jazz over the Spector is a slightly thinner, possibly easier to play neck.

Has anyone played both? Which would you recommend? Does the MIM Fenders have noticebly worse construction/sound quality than an American?

I've played both.

The MIM Fender is great. It needs a few...adjustments, I'd say. A new bridge mostly. It is noticably different to an American, I own an American and it's just such a difference compared to the Mexicans, but the Mexicans are still worth it.

The Spector I did like, but I preferred the Shecter in it's price range. I dunno, the lower end Spector's just don't feel...right to me. Maybe I'm justnot a fan of them themselves.

I would myself, probably go for the MIM. They're fantastic upgrading basses, and they're a step up from starter basses to the point where you can comfortably play it in a gig. The main thing you need to do is play both at a store before buying, and decide the which you prefer. Spectors have a very unique feel imo, so, you need to see if that suits you before you buy.
You're comparing two VERY different basses. The Spector is a very hot, active bass (I think), and it's definitely more geared toward modern rock, metal, even pop to an extent. The Jazz is geared toward versatility. Doing well in everything, and doing great in classic rock, blues, that kind of thing. Play them both and decide for yourself. The difference between MIM and MIA is noticeable but you also have to take into account the MIM is half the price. It is not half the quality. But, it is definitely not as good as an MIA, the MIA's are far superior in tone, quality and feel. But, again, the MIA's are far superior in price tag as well. For the price, most would agree the MIM's are well worth it.