What is empirical entropy? In the definition of jointly typical sets (in "Elements of Information Theory", ch. 7.6, p. 195), we use 
$$-\frac{1}{n} \log{p(x^n)}$$ as the empirical entropy of an $n$-sequence with $p(x^n) = \prod_{i=1}^{n}{p(x_i)}$. I never came across this terminology before. It is not defined explicitly anywhere according to the index of the book.
My question basically is: Why is empirical entropy not $-\sum_{x}{\hat p (x) \log(\hat p(x))}$ where $\hat p(x)$ is the empirical distribution?
What are the most interesting differences and similarities between these two formulas? (in terms of properties they share/do not share). 
 A: Entropy is defined for probability distributions. When you do not have one, but only data, and plug in a naive estimator of the probability distribution, you get empirical entropy. This is easiest for discrete (multinomial) distributions, as shown in another answer, but can also be done for other distributions by binning, etc. 
A problem with empirical entropy is that it is biased for small samples. The naive estimate of the probability distribution shows extra variation due to sampling noise. Of course one can use a better estimator, e.g., a suitable prior for the multinomial parameters, but getting it really unbiased is not easy. 
The above applies to conditional distributions as well. In addition, everything is relative to binning (or kernelization), so you actually have a kind of differential entropy. 
A: If the data is $x^n = x_1 \ldots x_n$, that is, an $n$-sequence from a sample space $\mathcal{X}$, the empirical point probabilities are 
$$\hat{p}(x) = \frac{1}{n}|\{ i \mid x_i = x\}| = \frac{1}{n} \sum_{i=1}^n \delta_x(x_i)$$
for $x \in \mathcal{X}$. Here $\delta_x(x_i)$ is one if $x_i = x$ and zero otherwise. That is, $\hat{p}(x)$ is the relative frequency of $x$ in the observed sequence. The entropy of the probability distribution given by the empirical point probabilities is
$$H(\hat{p}) = - \sum_{x \in \mathcal{X}} \hat{p}(x) \log \hat{p}(x) = - \sum_{x \in \mathcal{X}} \frac{1}{n} \sum_{i=1}^n \delta_x(x_i) \log \hat{p}(x) = -\frac{1}{n} \sum_{i=1}^n \log\hat{p}(x_i).$$
The latter identity follows by interchanging the two sums and noting that $$\sum_{x \in \mathcal{X}} \delta_x(x_i) \log\hat{p}(x) = \log\hat{p}(x_i).$$
From this we see that 
$$H(\hat{p}) = - \frac{1}{n} \log \hat{p}(x^n)$$
with $\hat{p}(x^n) = \prod_{i=1}^n \hat{p}(x_i)$ and using the terminology from the question this is the empirical entropy of the empirical probability distribution. As pointed out by @cardinal in a comment, $- \frac{1}{n} \log p(x^n)$ is the empirical entropy of a given probability distribution with point probabilities $p$. 
