Stone–Čech compactification vs Stone Duality

Question

Let $B$ be a Boolean algebra. There are two ways to turn $B$ into a topological space.

1. View $B$ as a discrete space and take the Stone-Cech compactification resulting in $\beta(X)$. This is the same as the Stone space associated to the Boolean algebra $\mathcal{P}(X)$ of powersets of $X$.
2. Take the Stone space of $\mathcal{B}$, $\mathrm{Ult}(B)$.

What is the relationship between $\mathrm{Ult}(B)$ and $\beta(X)$?