What does the statement of cash flows tell me that the income statement can't

The income statement has revenues and expenses on it, done on an accrued basis. A cash flow concentrates on just cash. They don't coincide. The fact that an income statement is (usually) done on an accrued basis means that revenues and expenses don't match cash. (You put something into receivables, so you've recorded the sales, but do you have the cash?) Also, there are things that are simply non-cash items on an income statement, like depreciation. There was no cash paid for depreciation. And there are many cash transactions that never show up on an income statement at all, like purchasing/selling plant assets, or debt transactions, or the selling of stock, etc. And some of those types of things can involve very large amounts of money. So there can, in fact, be a pretty big difference between what shows up on that income statement and what is showing up on the cash flow.

Simple go find an example of each and it becomes easy to see some of the differences, at least the more obvious ones.