Why is mutual information symmetric but conditional entropy isn't?
No,because while mutual information tells you how much uncertainty can a random variable $X$ remove from another random variable $Y$, the conditional entropy of $X$ given $Y$ tells you how much uncertainty remained in X after using the information (given by $I(X;Y)$) that $Y$ gave it.

$H(X|Y)=H(X)-I(X;Y)$

That's why the conditional entropy depends on the value of the entropy before the observation and the mutual information isn't, because it is only the difference ($\delta$) between two entropy states, before and after the observation.