It depends on the car (both the "old" one and the "new" one), but if you're getting an efficient car then you'll be saving on tax and fuel. Ongoing maintenance, all going well, there probably won't be much difference. However, you're less likely to get hit with an unexpected with a new car, and if you do it'll be warranty covered. If our 10 year old can limp through to August we'd be looking at a day to day saving. On fuel alone, the costs per week will be at least halved by my calculations.
The caveat I would throw into the calculations is that it'd be changing from a family saloon to a hatchback, so the comparison probably isn't totally fair. But then I think this is what a lot of people are doing, particularly in 2 car households. People can no longer justify 2 family size cars when one of them only has just the driver in it.