These are different designs, developed at different times. Forty years have gone between each was initially launched as the A-mount was simply acquired from Minolta which had by then fused into Konica-Minolta.

The A-mount introduces AF which worked by Phase-Detection and hence lenses for that mount are designed to focus that way. Over the years, they were optimized to provide better performance with Phase-Detection, moving the lens element a set distance faster.

The E-mount is all electronic and is built for lenses which have their own stepping motors to move the lens efficiently in tiny increments which is needed for Contrast-Detect AF. Of course, since a few NEX cameras offer Phase-Detect AF too, the new E-mount lenses are designed to work with both.

Optically the E-mount requires a much shorter flange distance which too small to accomodate the mirror needed by a DSLR and most-likely an SLT. This is why no DSLR uses the E-mount. Mirrorless cameras can use both since one can bridge the gap of flange distances by adding a simple tube with pass-through electrical contacts.

Sony has such adapters which some in two versions. One for manual focus, perhaps supporting SAM lenses (someone can correct this if this is wrong) and the other for autotocus. Those have a pelicle mirror and Phase-Detect AF built-in. Additionally, both these come in APS-C and Full-Frame variants but they respective advantages are the same.