Well, more often than not with scrappage schemes you get more (equivalent) money for your old "banger" than what the car itself is worth. So you basically acquire a, say, 10K asset for 8K, so you're quids in to start with. Considering that automakers usually add their own incentives on top of the government one, and that a 10 year old car is likely to be less efficient and more prone to breakdowns than a brand new one, you're likely to save on upkeep, fuel, tax and insurance (newer cars, for example, are (usually) more difficult to hot-wire than old ones so harder to steal). It depends on the car you trade in and the one you acquire, of course. With 'eco' cars being all the rage at the moment, the savings might be very interesting.
For old cars (2001-) the tax is linked to displacement (my bad) but a new one is still potentially more economical. A Vauxhall Corsa base model from the year 2000 pays 125 a year. A 2010 Corsa base model pays zilch the first year and 30 quid per year afterwards. Over 10 years you save 920 squids in tax if you buy the new Corsa than if you keep the old one. Plus you don't have to MOT for the first three years (at ~50 a pop), fuel consumption will probably be more efficient especially if you buy an 'eco' model and you'll enjoy all the benefits of owning a new motor (warranty, lower insurance, reliability etc.).