According to the translator, 編修 and 編集 are pronounced as "Henshū" and given the same meaning of "Edit". Also, both can be used as a noun.
So are there any specific situations to use these words?
There is some degree of overlap, but here is the basic difference.
編集 is a very common word that means "edit(ing)". In the publishing industry, 編集 mainly involves DTP works, typo correction, terminology standardization, cover design, creating an index, and so on. A professional 編集者 (editor) working at an editorial office of a publishing company also does scheduling, planning, budgeting, calling for articles, etc. 編集 also means editing of a video file, programming source code, etc.
編修 is a relatively uncommon term which is more like "compilation". It's about gathering materials from various sources and making a large work such as anthology, encyclopedia, and chronology. A person who does 編修 is not a DTP operator but an expert of a certain academic field. 編纂 is a more common and less confusing synonym. Admittedly, a person who does 編修 also does 編集 to some degree, so the borderline is sometimes blurry. 辞書の編集 and 辞書の編修 seem to be used almost interchangeably.
For example, an odrinary novel doesn't need 編修, but every book needs 編集. When in doubt, you can just think "editing" normally corresponds to 編集.