First, here's the real logic behind twin studies (not the bastardized version given in the Slate article). Siblings and twins are more genetically similar than two randomly chosen members of a population. Here's where the confusion typically begins. As a shorthand, people often say, "siblings and dizygotic (fraternal) twins share 50% of their genes." That's sort of true, but also misleading, because all humans have about 99.9% of their genomes in common. Out of about 3 billion base pairs, two people typically vary by a few million only. So siblings clearly share more than 50% of their genomes. A more precise statement about siblings would be "siblings and fraternal twins, on average, have only 50% of the genetic variance of two randomly chosen people."
Monozygotic (identical) twins, on the other hand, are even more genetically similar than dizygotic twins or siblings. For the moment let's assume they are genetically identical, even though it turns out that's not really true. Many studies then look at sets of fraternal vs. identical twins, who encounter similar amounts of environmental variation within pairs. That is, the differences in the experiences of two fraternal same-sex twins is likely to be similar to the difference in the experience of two identical twins. Then when they're adults, you measure them for any number of behavior traits--height, weight, political affiliation, whether they prefer calling or texting. The members of the sample then give you a range of responses, or data points. Take height. In a typical sample of 100 males, you might find a range of height from 5'3''-6'5'', or whatever. If you choose two people randomly from this sample, the likelihood that they are close in height is a random function of the sample itself. But if you choose two fraternal twins, you find a correlation: fraternal twins are much more likely to be close in height than two random people. And if you choose two identical twins, you find an even better correlation.
Conclusion. Genes determine height. JUST KIDDING. That's the straw man version of the conclusion given by Palmer, which no one who really knows biology would ever claim. The conclusion is that a certain proportion of the variance in height (best estimate: ~50%) across a population sample is correlated with genetic variation. This is very different from the idea that "your genes determine 50% of your height." That statement, about an individual, is utterly incoherent. An old psychology textbook I read used the analogy of a rectangle, with width being its metaphorical genes and area being its metaphorical traits or behavior. The latter statement would be equivalent to saying "a rectangle's area is determined 50% by its width." But that simply doesn't make any sense, because area and width are measured in different units, and the area is entirely determined by the height and width together. What you can do is compare two (or more) rectangles and say "50% of the difference in their areas is due to the differences in their widths." That's what geneticists mean when they a trait is 50% heritable.
So when Brian Palmer says "genes determine half your altruism," or "one quarter of your financial decision," he's seriously mischaracterizing, (in the name of simplifying?), what some study found about altruism and financial management. Genes don't make your financial decisions for you, genes interacting with an environment create a person who makes those decisions. But like the rectangles, differences in genes can account for measurable differences in behavior across a given population, including your behavior with respect to financial decisions. This is not at all surprising.
What's new to me is this idea that identical twins' genomes are actually quite different, and I'd have to read more about it to fully understand "copy number variations." But unfortunately for Palmer, I don't think this fact actually helps his argument at all; if anything, it hurts it! Consider: you take a sample of identical twins (let's say separated at birth so we can ignore the effects of parenting or "shared environment, although *new tab* these effects turn out to be quite small), and a control sample, and find that, for a given trait or behavior (let's stick with height to keep it simple), the twins' scores correlate 50% better than the control sample. That is, identical twins are 50% more similar in height than two random people. Your initial conclusion: height, in this population, is 50% heritable. But wait! It turns out, identical twins aren't genetically identical at all, they in fact differ genetically quite a bit. Does that mean we should downgrade the heritability we estimated for height? No, in fact, it means the heritability is in fact higher. Each twin's environment varies, on average, as much as the people's environments taken in the random sample. That means the 50% correlation coefficient is best explained by genetic similarity (or more precisely, a lack of genetic variation) among the sets of twins. But if that genetic similarity isn't 100%, but lower, then less genetic similarity has to account for more behavior similarity, which means the behavior similarity to genetic similarity ratio has gone up, and genes are more important than the initial conlcusion suggested.
Heritability is something that can only measured in a given population, and genes never "determine" any portion of behavior in an individual. Palmer's other points are largely irrelevant to the value of twin studies, and population genetics more generally.