When you publish in CATIA, you are "telling the world" (or at least your co-workers) which features of your design are to be used on other parts or assemblies.
These features might be geometric elements (points, lines, surfaces, etc.) that are to be used as "master geometry" and will be linked to children parts. Or, they might be common parameters that are to be linked and used with several parts.
These features might also be PartBodies that are linked to children parts. (for example; a casting that is copied with a link to a second part where the machining operations are added)
Publications might also include design features that are to be used for Assembly Constraints.
In my opinion, there are two major benefits to Publishing:
1. By Publishing (and naming) certain features, those featues will automatically be replaced whenever a parent part or master geometry is replaced. If features are not published, if the parent part is replaced (or renamed) the result will be broken links that must be replaced manually. With Assemblies, publications will result in fewer broken constraints as parts are revised.
2. Published featues can quickly be identified and consumed (or used) with confidence that things are being done as the creator of the master geometry intended. This is especially beneficial when the design is being done by a team of several/many people.
