Beta(20,10)

Where 20 is the number of failed attempts and 10 is the number of successful attempts (for a total of 30 attempts). Given this, I was successful on one-third of the attempts and indeed the mean of this distribution (as I've computed it) is 0.3333. However, the mode (as I've computed it) is 0.32142857142857145.

My question is, why is the mode (the value at which the PDF is maximized) less than the mean? If I represent this experiment with a binomial distribution Binomial(n = 30, p = 1/3), the mean and the mode are equal.

Conceptually, it's easier for me to understand the Binomial distribution as it seems really straight-forward: Out of 30 tries, I made 10 field goals. But with the Beta distribution, I'm not sure I really understand what it is saying, hence I don't understand why the mode and mean are different. I understand that the Binomial distribution is discrete and the Beta distribution is continuous, so I'm assuming that has something to do with it.....