PHOENIX boards have modified arcade program code. BATTERY boards have O.E.M. CAPCOM code.
And BATTERY boards also have suicide batteries that are known for killing arcade boards in various ways. But, this is the O.E.M. game code that you played in the arcade.
PHOENIX code is designed to revive dead arcade boards by modifying the O.E.M. code. It removes the suicide code (and suicide battery) from the whole program and adds a few extra fun programs.
Some gamers say they notice difference in game play on a PHOENIX board; however, other games say they do not: that PHEONIX boards play just a well as the BATTERY boards. I've only played on a BATTERY board; so, I can't say there's a difference.
IMHO, in general, if you have a BATTERY board, it's better to change it; and, if you have a dead board, then it's better to PHOENIX it. Except, if you have a LOT of BATTERY boards and you don't want to keep track of all the BATTERY CHANGE DATES, then just PHOENIX you battery boards whenever you get the time, it's a lot easier that way.