The two aren't the same.
This came up recently, when I was discussing the difference between the words "convince" and "persuade" with a co-worker. According to the Associated Press Stylebook, you convince someone that or of something. You persuade someone to do something. Convincing takes place in a conceptual realm. It doesn't lead to physical action, but mental acceptance. Persuading goes further. It involves pushing someone to take real, concrete steps.
Anyway, that's what AP and other style manuals say. I explained this. My co-worker then asked if that distinction really made sense, given that many use the words interchangeably. I hemmed and hawed, then acknowledged little difference exists between the words in common usage.
That's not the point, though. Editors follow many rules that don't necessarily make language more grammatical. Newspaper usage does not equal common usage.
The AP stylebook and similar reference works don't exist to teach newspaper copy editors grammar. Generally, these books assume familiarity with language. They exist to establish a standard of usage for serious news reporting. They exist so readers can know that important stories will be conveyed in clear, understandable prose, free from jargon or slang. They exist for the sake of consistency.
Copy editors should know style. They should, most of the time, follow it. But they should also be aware when it diverges from common usage, and why. To my mind, the "convince/persuade" distinction still has value. The words still suggest slightly different concepts.
But as time passes, and common usage flattens the differences further, perhaps AP editors will remove the style rule. It won't be the first time.