Newcastle will always be cheaper housing-wise. The cost of living is much better up north. However, for settling down I would think about what your priorities are. For me, work opportunities are more important than cost of living - otherwise I'd live somewhere like Lincoln, Scunthorpe, Dunfermline, or some dead town where rent is dirt cheap (but at the cost of relatively few work opportunities).

If you were to relocate to Leeds, consider this: from Leeds, you could regularly commute into Manchester and Sheffield as well, which gives you the work opportunities of three fairly large cities. Whereas with Newcastle, you will only also have commutable opportunies in Sunderland and Middlesbrough.

These two places have less than 500,000 people between them, compared to the over 1 million living in Manchester and Sheffield. Not only does Leeds offer more opportunities due to there being far more people in a commutable range, but there are also way more companies (and companies of different types!) based in Manchester, which is ultimately both England and the UK's third largest city (+ de facto capital of the North).

So if youre thinking about settling down for the long run, I'd personally choose Leeds only because there are greater and more numerous commutable work opportunities should you ever lose your job, be made redundant, or desire a career change. There are also more further and higher education opportunities around, and so if your children go to uni I'm sure you'd ideally want them to be only 1-2 hours away if they were ever in an emergency.