An abbey simply refers to a society of people, secluded from the world and devoted to different religious activities, which is headed by an abbess (the spiritual mother) and abbot (the spiritual father). An abbey can as well be referred to as a monastery where religious people dwell, but it is bigger than a monastery since it contains different infrastructures that are used for different purposes.
For instance, an abbess consists of buildings meant for activities like counseling, praying, grooming, etc. On the other hand, a cathedral is most times referred to as a church because, just like for most churches, the main reason behind the creation of a cathedral is to serve as a place of worship.
However, a cathedral, in its actual meaning, is a principal church which serves as the central church of a diocese. While abbeys are headed by abbesses and abbots, cathedrals are headed by bishops.