There are a few issues with BC/AD:
A.D. 1 was first calculated in the first millenium based on available knowledge at the time. Later on, it was found Jesus likely wasn't born that year, but a few years earlier (i.e., in the somewhat ironic 3–4 B.C. area). Marking it as the "Christian Era" (or more commonly, the "Common Era") allows the same epoch to be used even though the best calculation for Jesus's birth has changed.
While Christians make up a very large chunk of the world's population, they are no where near the majority. Most organizations and political entities, for the sake of convenience, have adopted the Western calendar, but "Anno Domini"/"Before Christ" are meaningless terms. Replacing it with "Common Era"/"Before Common Era" reinforces the notion of a global, common epoch starting at the height of the Roman Empire.
When "Christian Era" is used, it's still clear what epoch is being referred to (i.e. the Western one) without having to have some special knowledge about what "anno domini" means or who Christ is.
Wikipedia also mentions an issue with the Julian Calendar and the Gregorian Calendar historically both using AD/BC, leading to some confusion as to which calendar system is being referred to:
The terms "Common Era", "Anno Domini", "Before the Common Era" and "Before Christ" can be applied to dates that rely on either the Julian calendar or the Gregorian calendar. Modern dates are understood in the Western world to be in the Gregorian calendar, but for older dates writers should specify the calendar used. Dates in the Gregorian calendar in the Western world have always used the era designated in English as Anno Domini or Common Era, but over the millennia a wide variety of eras have been used with the Julian calendar.
Switching to CE/BCE makes it clear the Gregorian calendar is being used.