Why isn't conditional probability sufficient to describe causality? I read these comments from Judea Pearl saying we don't have causality, physical equations are symmetric, etc. But the conditional probability is clearly not symmetric and captures directed relationships. 

How would Pearl respond to someone saying that conditional probability already captures all we need to show causal relationships?
Perhaps the shortest answer to this question is that Bayes' Theorem itself allows us to easily change the direction of a conditional probability:

$$
P(A|B) = \frac{P(B|A)P(A)}{P(B)}
$$

So if you have $P(B|A)$, $P(A)$, and $P(B)$, we can determine $P(A|B)$, and similarly you can determine $P(B|A)$ from $P(A|B)$, $P(B)$ and $P(A)$. Just by looking at $P(B|A)$ and $P(A|B)$, it is therefore impossible to tell what the causal direction is (if any).

In fact, probabilistic inference usually works the other way round: When there is a known causal relation, say from diseases $A$ to symptoms $B$, we usually have $P(B|A)$, and are interested in the diagnostic reasoning task of determining $P(A|B)$ from that. (The only other thing we need for that is the prior probability $P(A)$ since $P(B)$ is just a normalization factor.)