Погибать is to perish. So it is more about violant death in the war or from some external causes like car accident or natural disaster, rather than just a death due to age. Мой дед погиб на войне. Вся семья погибла в авиакатастрофе. От падения крана на жилой дом в восточном Лондоне погибли люди. You can say it about any living being, e. g. "У меня твой цветок погиб", which means the flower you gave me died, because I watered it too little or too much, our just left it in a dark cold shed and forgot about it, or the conditions at my place were just wrong for it. Гибнуть is the same. But it doesn't have a perfective form. So it is used more often to reflect continuous action in the past, present and future like "мы тут объедаемся каждый день, а где-то в Африке люди гибнут от голода", or to say that something used to die regularly, like "в каждую суровую зиму молодые телята не доживали до весны и гибли от холодов". But the difference between two verbs is very small.

They mean the same, only the form is different.