'Vibe' is an informal word that you wouldn't use in formal contexts. 'Atmosphere' is a neutral word that can be used in any context. In some cases, they can mean the same thing. You can say that a bar has a 'good vibe' or a 'good atmosphere', referring to the general ambiance of the place. 'Vibe', which comes from the word 'vibration', can also mean a general impression or feeling received. For example, if you say 'I'm getting some bad vibes about this guy', it means that you instinctively feel there's something not right about him. You can't use 'atmosphere' in this way. 'Atmosphere' can be used in many different situations to describe non-tangible aspects of one's surroundings - 'a friendly atmosphere' 'a cosmopolitan atmosphere' 'a studious atmosphere' and so on. It also has a meaning in science, for the gases etc, surrounding a planet.

They are the same. Vibe is more informal.