For Italy, you must know that the Antenati and FamilySearch databases are not identical, they overlap in parts but not as a whole.
Antenati has civil records for the most part, those indexed can be find through FamilySearch too, but not those that are not indexed yet. Marriage "processetti" (folder with the documentation they had to present prior to getting married) are specially useful though not indexed. Many times they have info about the parents of the groom and bride, previous marriages of one or both of them and children born before the marriage that are not reflected in the marriage record itself.
On the other hand, if you search in FamilySearch through the "Catalogue" for a certain city/town/place in Italy, you may get lucky and find Catholic Church records, not available elsewhere. Also some census records. Neither are indexed for the most part, but it's better than nothing. usually they're very easy to read.
A general tip for research, since my family has branches in many countries (and languages), when I get stuck or blocked or confused with one, I jump to another for a while, then when I come back to the first one after a while I recheck all the info I have and many times I immediately notice a detail that leads to new discoveries, something that I may have overlooked earlier because my brain was too tired or overloaded with data and just going around in circles.
Also something that has helped me a lot with research (specially in Europe with all its dramatic geopolitical changes in the past centuries) is to learn a bit about the history of the particular town or place I'm researching (through Wikipedia usually - sometimes the names of the places change over time) and to check historical maps like the ones you can find here:
https://maps.arcanum.com/en. The overlapping feature between the historical map and the modern GoogleMaps area is awesome!
I hope this helps!