A.D./B.C. or C.E./B.C.E.?

It is becoming increasingly common to find dates that have for centuries been described as, for example, "A.D. 1776" or "44 B.C." are now being described as "1776 C.E." or "44 B.C.E.".  Does this new usage demonstrate a greater sensitivity and a more careful, academic approach?

No.  To be perfectly blunt, it is neither more nor less than an anti-Christian bias self-righteously masquerading as intellectualism. 

That is a serious charge, but the facts back it up.  Advocates of the new terminology say that the older terminology is offensive to those who do not accept Christianity.  However,

1. These same people make no attempt to rename the months of the year, many of which are named after Greco-Roman gods.  How is it that Christianity gives toxic offense, but not Greco-Roman paganism?
2. The same is true of days of the week:  in English, they are named after Germanic gods.  Again, the people who find Christ offensive seem to think Thor and Wotan are perfectly acceptable. 
3. Pay close attention to how foreign words and names are pronounced by the "C.E." crowd.  You will find that they go to great lengths to give the native pronunciations, or at least what they think are the native pronunciations.  They often over-correct for Anglicization, but they want it to be obvious that they are so solicitous for cultural sensitivity towards speakers of Arabic, or Hindi, or Japanese, or Mandarin Chinese.  They would not dream of behaving like a European colonialist and appropriating someone else's cultural artifact and then obscuring its origins.  With a few exceptions, that is -- because that is exactly what they are doing when they take the A.D./B.C. calendar system, which is a cultural artifact of Christianity, then try to obscure its Christian origins.  This inconsistency is part of what makes this not merely anti-Christian, but also pseudo-intellectual.
4. The other thing that makes it only pseudo-intellectual is the contrast with the metric system that is used in all modern science.
   • For one thing, when the French Revolutionaries decided that every vestige of the French monarchy was offensive, they did not simply rename the toise the metre, rename the pinte the litre, and rename the denier scruple the gram.  They developed a completely new system that was intended to be both practical and as nearly universal as possible.  So, for example, the meter was defined to be one ten-millionth of the distance between the north pole and the equator, the liter was defined to be one thousandth of a cubic meter, and the gram was defined to be one thousandth of the mass of a liter of water.  Because these units do not show favoritism or hostility to any people or culture, they have been universally adopted in the sciences.
   • As more and more precise measurements have become necessary, the metric units has been slightly altered so that they are defined in terms of experiments that yield consistent values at higher precision.  For example, the second is now defined not by the rotation of the earth, but from the frequency of light emitted by a particular transition in cesium atoms, resulting in more precise and accurate measurement of time.

At least the last two points can be fixed.  If it is decided to move away from the A.D./B.C. system for academic work, my suggestion would be to set year zero to A.D. 775, when intense solar storms left a distinctive mark in carbon-14 dating.  It has been suggested that this event could be used to calibrate tree ring dates from around the world, leading to much more precise dating of events in the ancient world.  This also has the advantage of corresponding to a world-wide physical event, so it is independent of culture and religion.  Also, for simplicity of calculations the years before A.D. 775 should be given negative numbers.  If this system were named the way SI (metric) units are named, it might be named after Andrew E. Douglass, the "father of dendrochronology".