Theorem. The Stone-Čech remainder of the real line contains a family of $2^\mathfrak c$ topologically distinct continua.
Here a continuum is defined to be a compact connected Hausdorff space. $2^\mathfrak c$ is easily seen to be an upper bound in the problem.
The proof was divided into two cases:
Case 1: The Continuum Hypothesis fails.
Dow, Alan, Some set-theory, Stone-Čech, and $F$-spaces, Topology Appl. 158, No. 14, 1749-1755 (2011).
Case 2: The Continuum Hypothesis holds.
Dow, Alan; Hart, Klaas Pieter, On subcontinua and continuous images of $\beta \mathbb{R} \setminus \mathbb{R}$, Topology Appl. 195, 93-106 (2015).
The proofs are radically different. In fact, the continua constructed for Case 1 are all homeomorphic under CH, and CH is essential to the constructions in Case 2.
This is the only theorem I know of which was proved using CH in this way.