To my opinion, 陪伴usually focuses on the more sentimental side, like 家人的陪伴,朋友的陪伴 the accompany of family/friends, While 陪同 is often used in a more formal way, for instance in newspaper"the president was accompanied by...."总统在....的陪同下...., or in a regulation “the child must be accompanied by his or her parents”孩子必须有家长陪同.
Hope I can be of help. ^_^