The simplest reason for using BCE/CE as opposed to AD/BC is to avoid reference to Christianity and, in particular, to avoid naming Christ as Lord (BC/AD: Before Christ/In the year of our Lord).
Why do we no longer use BC and AD?
There is no biblical authority for BC/AD; it was created over 500 years after the events described in the Christian New Testament. Dionysius never makes the claim that he knew the date of Jesus birth and no later writer makes that claim for him.
What do we use now instead of BC and AD?
Breaking with centuries of tradition, the terms BC and AD are to be replaced with a system known as the Common Era. The Latin term Anno Domini, meaning in the year of our Lord, becomes Common Era, or CE, and Before Christ becomes Before the Common Era, or BCE.
Do we still use AD and BC?
The movement towards BCE/CE has not been universally accepted, and BC/AD is still more widely used, even though BCE/CE has been in the mainstream since the 1980s.
What do we say now instead of BC?
Common Era (CE) is one of the year notations used for the Gregorian calendar (and its predecessor, the Julian calendar), the worlds most widely used calendar era. Before the Common Era (BCE) is the era before CE. BCE and CE are alternatives to the Dionysian BC and AD notations, respectively.