What are the advantages of filing a priority patent in one country before filing a patent for that same invention in a number of other countries?

You don't actually file a "priority" patent (application) in one country before filing a patent application in another country - the patent application that you file in your first country automatically becomes your priority patent application. Many/most countries require the filing of a patent application in the country where the invention was first developed before they will permit patent applications to be filed in foreign countries, and then once the first patent application is filed, you have one year to file in foreign countries or PCT in order to claim priority back to that first patent application.