Step 1: Compare the two conjugate bases side by side, since the stronger acid is the one whose conjugate base can better accommodate the extra negative charge.
Step 2: Removing the proton from phenol gives phenoxide, where the charge is not stuck on oxygen but is shared around the aromatic ring by resonance. Several contributing structures place the negative charge on ring carbons, so the ion is low in energy and stable.
Step 3: Removing the proton from ethanol gives ethoxide, where the charge remains fully on one oxygen. The electron-donating ethyl group even increases the electron density there, so this anion is comparatively high in energy and unstable.
Step 4: A stable conjugate base means the forward ionisation is favoured. Since phenoxide is far more stabilised than ethoxide, phenol ionises to a greater extent in water, giving it a larger acid dissociation than ethanol.
Result: Phenol is the stronger (more acidic) of the two.