Skew and Kurtosis are so similar?
A simple and direct answer is that skewness and kurtosis are both defined in terms of the $Z-$values, $Z = (X-\mu)/\sigma$ : Skewness = $E(Z^3)$ and Kurtosis = $E(Z^4)$. When talking about a data set, you can just replace the expectation operator "$E$" with the ordinary average. 

Since raising a number to either the third or fourth power amplifies values that are greater than 1 (in absolute value) and diminishes values that are less than 1 (in absolute value), both skewness and kurtosis are primarily measures of the heaviness of the tails of the distribution of $X$. 

Note that the $Z$-values than are less than one in absolute value correspond to $X$-values that are less than one standard deviation from the mean; greater than one correspond to values that are more than one standard deviation from the mean. Hence, the averages of the $Z^3$ and $Z^4$ values are highly influenced by the $X$-values that are many standard deviations from the mean; i.e., by the tails or outliers.

Consider skewness first.  By definition of skewness, and by definition of center of gravity, the graph of the probability distribution of $Z^3$ balances (i.e., has a center of gravity) at the skewness of $X$. Hence, if the tails are heavier on the left side of the distribution than on the right, then skewness is negative. Conversely,  if the tails are heavier on the right side of the distribution than on the left, then skewness is positive.

Now consider kurtosis. The graph of the probability distribution of $Z^4$ balances (i.e., has a center of gravity) at the kurtosis of $X$. Note that the kurtosis of the normal distribution is exactly 3.0. Hence, if the tails are heavier on either or both sides of the distribution than the tails of the normal distribution, then the kurtosis is greater than 3.0, and the probability distribution of $Z^4$ falls to the right when a fulcrum is placed at 3.0. Conversely, if the tails are lighter on either or both sides of the distribution than the tails of the normal distribution, then the kurtosis is less than 3.0, and the probability distribution of $Z^4$ falls to the left when a fulcrum is placed at 3.0

Hence, skewness and kurtosis really do not measure things that differently - they are both measures of tail heaviness. The biggest difference is that skewness considers relative heaviness in one tail versus the other. The Wikipedia entry on kurtosis displays both lower and upper bounds for kurtosis in terms of skewness.   

There is a mistaken but remarkably persistent meme out there that kurtosis measures "peakedness or flatness" that might be a source of confusion.  Your work shows (yet again) that this characterization is false. See  here for a clear explanation of why the "peakedness/flatness" characterization is wrong.