There is this thing called "topology" that is used for this.
Basically, you
- Create a geodatabase.
- Create a new feature dataset in the geodatabase.
- Copy or move the polygons as a feature class inside the feature dataset.
- In that feature dataset, alongside the polygon feature class, create a new topology.
- Onclude the feature dataset in it.
- You then add rules for the dataset such as Must NotOverlap and Must not have gaps.
- Then you validate it.
- The result show you highlighted gaps and overlaps.
There are topology tools that allow you to decide what to do with the overlaps or gaps: that is to assign them to one side or the other of the overlap or gap.
Hope this makes sense and helps.