How many points does Stone-Čech compactification add?
There are spaces like $[0,\omega_1)$ with the order topology where the Stone-Cech compactification is the same as the one-point compactification.
On the other hand, the Stone-Cech compactification of the natural numbers $\mathbb N$ has cardinality $2^{\mathfrak c}$. So the answers depends a lot on the space.