For the space $\omega_1$ (with the order topology) we have $\beta\omega_1=\omega_1+1$ (or $\beta[0,\omega_1)=[0,\omega_1]$, if you prefer this notation), i.e., it is an example of a space for which Stone-Čech compactification and one-point compactification (a.k.a. Alexandroff compactification) coincide. (See, for example, this answer and this blog.)
Is there some known characterization of topological spaces such that Stone-Čech compactification $\beta X$ and one-point compactification $\omega X$ are the same?