with the hope of强调的是客观事实,eg.With the hope of recovery, she tried live optimistically.这里讲的是,因为有希望复原,她很乐观地生活.
而in the hope of,强调的是主语的主观愿望.She worked very hard in the hope of promotion.这里讲的是,她想得到提升,所以工作得很努力.