Phenol may be considered as a resonance hybrid of five cannonical forms, as a result the oxygen atom acquires a partial positive charge. This weakens the     O-H bond and thus facilitates the release of proton.

The phenoxide ion thus formed also exists as a resonance hybrid of five contributing structures. Phenoxide ion is better stabilized than phenol because in phenoxide ion only the negative charge is spread over the ring while in phenol three contributing structures involve charge separation( positive on oxygen and negative at ortho & para positions of ring).Hence phenol has tendency to form more stable phenoxide ion by releasing a proton.

However, in case of alcohol, both alcohol molecule & the alkoxide ion are represented by single structure and there is no resonance. Further, the alkoxide ion due to the presence of formal negative charge on it has greater energy and hence less stable than alcohol. Thus alcohols have negligible tendency to release a proton.