Lament and mourn are slightly different, even though they are listed as synonyms. Lament is used more for regret. "Many people lament the passing of old classical films into oblivion" Mourn is used more for sorrow at someone's death. "Her passing was mourned by all who knew her." If somebody was mourning rather than lamenting, I would probably be much more likely to see tears.

As synonyms they are interchangeable.