歇息 is slightly more colloquial / old-fashioned than 休息. It carries an extra 'lodge for the night / turn in' sense that 休息 lacks: 找个客栈歇息 = find an inn to put up for the night. In wuxia or period dramas, 歇息 is the default; in modern offices, 休息 wins.
memory
Both characters mean 'rest' on their own - 歇 (xiē, to stop and rest, as in 歇业 close shop) and 息 (xī, breath / rest, as in 休息). The pair stacks two synonyms for emphasis, a common Chinese disyllable strategy.
