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What is the difference between correlated equilibrium and mixed equilibrium?

Here's what I understand :

Unlike a pure Nash equilibrium, a mixed equilibrium corresponds to when each player has a probability distribution he follows instead of pure actions (single action with probability one).

Also, what I understand by correlated equilibrium is that it corresponds to when someone tells each player to follow some particular probability distribution, and each player plays according to that.

What I don't understand is what is the difference between these two? Can't one be modeled as the other?

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In a mixed (strategy) Nash equilibrium, the players' actions (strategies) are independent random variables. In other words, if you know that player 1 (randomly) chose $x$, that doesn't give you any additional information about what player 2 might do.

In correlated equilibria the actions (strategies) need not be independent. Wikipedia gives an example in the setting of the game of chicken, showing that the expected value of a correlated equilibrium can be greater than that of the mixed strategy equilibrium.

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    Just one more question. This means that a mixed equilibrium is also a correlated equilibrium, right? (And not the other way around). – 2017-02-13
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    @Ojas Yes, a mixed strategy Nash equilibrium is a correlated equilibrium. In particular, all Nash equilibria (pure or mixed) are (possibly degenerate) correlated equilibria but not vice-versa. (Pure strategy Nash equilibria are degenerate mixed strategy Nash equilibria.) – 2017-02-14