What is the difference between "auto da fe" and "bonfire of the vanities"?

The difference between an Auto de Fe and the Bonfire of the Vanities is the Bonfire was an actual bonfire for the purpose of burning objects deemed sinful, while the Auto de Fe (Act of Faith) is the ritual of public penance of condemned heretics and apostates that took place when the Spanish or Portuguese Inquisitions had decided their punishments.

The bonfires
Savonarola started to encourage his followers to destroy anything which could be considered luxuries - books, works of art, musical instruments, jewellery, silks and manuscripts were burnt during the period of carnival around Shrove Tuesday.

These events became known as the 'bonfire of the vanities': the biggest of these happened on 7 February 1497, when more than one thousand children scoured the city for luxuries to be burned. The items were thrown on to a huge fire while women, crowned with olive branches, danced around it.

Such was Savonarola's influence that he even managed to get contemporary Florentine artists like Sandro Botticelli and Lorenzo di Credi to destroy some of their own works on the bonfires. Anyone who tried to resist was set upon by Savonarola's ardent supporters, known as piagnoni (weepers).

https://www.historyhit.com/1497-bonfire-vanities/

The phrase auto de fe (Act of Faith) refers to the ritual of public penance of condemned heretics and apostates that took place when the Spanish Inquisition or the Portuguese Inquisition had decided their punishment (that is, after the trial). The phrase also commonly occurs in English in its Portuguese form auto da fé.

https://www.jewishvirtuallibrary.org/auto-de-fe