When does SSIS SQL Server Configuration write to the configuration table?
The only actions that will cause the values in your configuration table to be updated within SSIS will be when you are in the Configuration editor menu. Clicking OK there will cause the values to be written to the table. 

Now, if you have an Execute SQL Task that has UPDATE statement in it, that would obviously change the values in the table but those would only become effective for the next run. Once a package is actually running, the configuration table could be dropped as it doesn't periodically poll the table to investigate values.

If you modify the value of a Connection Manager in your package, assuming that CM is also under Configuration, the next time the package attempts to validate itself at the package level, it'll reach out to the configuration resource (table, XML, etc) and overlay with the configured values. I think I remember that if you are just right clicking and executing a task within a package, it doesn't make the hop but don't quote me on that.