As visualized in the picture as an example, two polygons should be aligned to each other. The boundaries of the green polygon should not be changed. The blue polygon may therefore only be adapted to the boundary of the green polygon.
The aim is to eliminate the gaps and overlaps.
I already tried Align under ArcMap. Unfortunately, not all overlaps and gaps are removed, because the adjustment is probably made on the basis of the vertices.
The manual search for errors via topology tools would be too time-consuming, since there are many boundary problems.
Best Answer
Using the Reshape Feature Tool is a little time consuming, but may provide a solution. Delete all the vertices within the incorrect feature class that are near the correct boundary. Using the Erase tool or holding the Shift key while selecting vertices may assist in deleting abundant vertices.
Then click the Reshape Feature tool in the Editor toolbar.
Then use the Trace tool
in combination with the Straight Segment tool 
to create new vertices along the correct boundary. As shown below, the traced vertices replace the deleted vertices.
Then Finish Sketch and it should appear similar to below.