Abatement vs. exemption
An exemption removes qualifying value from taxation while an abatement generally reduces or phases the taxable obligation under a specific legal program.
New Jersey example
The words are often used loosely in marketing; the actual ordinance, approval and tax treatment control.
Official source
https://www.nj.gov/treasury/taxation/lpt/localtax.shtml
This is a plain-English summary. Rules and dollar amounts change, so check the official source or a professional before you file or sign anything.