Not the guy you asked, but let me try to explain the difference as far as I understand it.

Filipino is the national language of the Philippines, based on the Tagalog language, with vocabulary from various different Philippine languages such as Cebuano, Bicol, Kapangpangan, Ilocano, among others. At least, that's how it is defined. However, since Filipino is so heavily based on Tagalog, the two names are used interchangeably to refer to what many people see as the same language.

Now, I am supposing that ‘provincial language Tagalog’ refers to the various dialects of Tagalog spoken outside of (Metro) Manila such as Bulacan Tagalog, Batangas Tagalog, Quezon Tagalog, etc.

As far as I know, the standard being taught to non-native students of Tagalog is taken from the Manila dialect.