First we should be clear about what organic means.
Regarding produce:
Contrary to what most people believe, "organic" does not automatically mean "pesticide-free" or "chemical-free" ... it means that these pesticides, if used, must be derived from natural sources, not synthetically manufactured.
Regarding Farmed Meats (some of the key requirements):
- Must be raised organically on certified organic land
- Must be fed certified organic feed
- No antibiotics or added growth hormones are allowed*
- Must have outdoor access
Organic food covers a wide variety of techniques and different types of foods that all have their own types of legal loop-holes, or possible work-arounds strategies.
There are many pesticides that are naturally produced by plants which farmers use, but that doesn't mean they aren't dangerous. In contrast, just because a pesticide is synthetic doesn't immediately indicate that it's more dangerous than natural products. Some pesticides that are approved by the government may be harmless in small quantities, but when organic producers attempt to refuse these, they may turn to alternatives that are even more dangerous, just to keep the label "organic".
In another case, farmers could claim "natural methods are insufficient to address critical issues of production", and then treat their animals with antibiotics while still calling the meat organic.
On the other side of this issue, some people argue that the less chemicals we consume in general, the better. Even if they might be government approved, that doesn't prove they're completely non-toxic.
The Stanford paper claim that there just isn't enough evidence to prove a significant difference in health benefits/risks, but it does suggest organic foods contain less pesticides. The health risks of consuming less pesticides though, are still contested.
"Healthier" depends on so many factors: How much pesticide is being used? How much exposure becomes a detriment? Does it have long-term consequences? (This question is still unknown and mostly unsupported by research. The best claims researchers can make is that organic foods may lead to less intake of pesticides.) What are the loopholes that producers are using?
So when you say "organic" and "conventional" it really depends on the process, producer, and product.