Organic farming does not mean farming without weedkillers or pesticides. It means farming with weedkillers and pesticides that are largely unstudied, compared with known quantities like Monsanto's products (whose positive and negative effects are well documented and widely understood).
The 'organic' products may be better on balance, but we don't actually know that, because the body of knowledge isn't nearly as developed yet.
That's my layperson's understanding, anyway. I could be wrong.
It's entirely marketing, there is no actual data supporting the claim that it's healthier for you. It succeeds by the "appeal to nature" fallacy, which causes a bunch of problems and gets decidedly anti-science in a few ways.
Using only "natural" pesticides because they are "healthier". This is unsubstantiated, again it's just an appeal to nature. It can become problematic because natural pesticides often aren't as effective, thus certain farms/crops require using an excessive amount- way more than they would if they used the more effective synthetic pesticides. The synthetic pesticides are of course accused of being unhealthy (because unnatural)- though this is generally untrue. Furthermore, even the natural pesticides cause pollution problems, and overusing them isn't good.
Because the organic lobby succeeds by the appeal to nature, they are also anti-gmo. Which is a big problem.
While there are entirely organic farms, a lot of organic products are really not even what they advertise themselves to be, as many organic farmers don't actually run organic farms: they run normal farms with a section of it that adheres to organic standards. These standards are pretty arbitrary, except for the fact that organic farming is a significantly less efficient use of farmland, which is going to become a problem within the next century. We need to be making efficient use of our arable land, as we don't actually have much left on the planet unless we start turning national parks + protected areas into farmland.
Largely, organic farming is a faux luxury built on anti-scientific ideas that does nothing except waste valuable land resources, sold to relatively privileged people who are willing to pay a premium to feel better about what they eat.
The 'organic' products may be better on balance, but we don't actually know that, because the body of knowledge isn't nearly as developed yet.
That's my layperson's understanding, anyway. I could be wrong.