Azure - Connection String in Key Vault vs Application Settings
The benefits as I see them are the general reasons to use Azure Key Vault


Secrets are centrally stored with options for authorization, auditing, etc.
Azure can be scripted to update the secrets on a timer - so in case a connection string gets in to the wrong hands, it is only valid for a day (or a week, or whatever)
In case the secret is shared between multiple applications, you only have to update it in one place (not the greatest benefit, but still nice)