The higher dose really does work stronger - 600mg ibuprofen measurably beats the 400mg when it comes to pain relief, especially if the headache is more severe. That's not just your mother's imagination, the difference is real, even if it might not be dramatically huge. But here's the important question: how often do you take the 400mg and doesn't a single tablet help you properly? If the 400mg relieves you enough, you don't need to go higher.
With dosing though, don't just take whatever. The 600mg tablets are available over the counter at the pharmacy, but there are limits - no more than 3 to 4 tablets a day, and not for longer than a couple of days in a row without talking to a doctor. Ibuprofen takes a toll on your stomach and kidneys over time, so especially if you get frequent headaches, it's better to check with the pharmacist about what makes sense. If the 400mg sometimes isn't enough, you could also try taking it earlier - right when you first feel it - that often works better than waiting until the pain really kicks in.