Both are verbs and both can often be used interchangeably. The slight nuance difference is that "unite" is mostly used for actual groups, as opposed to "unify", which is used more for the joining of ideas and concepts. So your example using Europe, can be made using both very naturally.

"uniting Europe" sounds to me like bringing the countries together for a common cause, like defending themselves or for political and economic advancement. "unifying Europe" sounds more like bringing the countries into a more coherent and consistent group by getting rid of undesirable variances between them (like different currencies and other institutions that make interaction and cooperation more difficult).