Imagine two situations. In one, you get a new job that pays an extra thousand dollars a year. In the other, you manage to spend about $20 a week less, saving a thousand dollars a year more than you had been.
In the first case, the thousand dollars is taxed (possibly at a higher rate than your existing salary.) If it's taxed at 30%, that's only another $700 in your pocket.
Money you spend, on the other hand, is post-tax. Any of it you can switch to savings is like giving yourself a raise bigger than the amount you save. And this is before taking interest into account.
There are a lot of ramifications to this simple fact. Are you considering a job that pays another couple thousand a year, but with a longer, more expensive commute? If the additional two grand is taxed at 30%, that's another $1400 in your pocket. If you end up spending an extra $30 a week getting to work, your raise has just cost you money.
If you can cut your spending, you've just given yourself a raise. And it might be bigger than you're imagining.