A little history first. The dreadnaught size was first produced by Martin for the Ditson company in 1919. HMS Dreadnaught was a British battleship in WWI and Martin liked the name for their largest guitar size. Martin branded Ds went into production around 1933. When Gibson introduced their answer to the big Martin in about 1934 they assumed the name dreadnaught was a trademark and called their shape "Advanced Jumbo". Gibson changed to a more Martin like square shouldered shape in the mid 60s. Early Martins had 12 fret necks and were longer and more rounded in the shoulders.
Personally I find Gibson AJs to be slightly more balanced and versatile than Martin Ds but it's close. Usually the Gibsons have a warmer midrange while the Martins are more "top and bottom" focused. This varies with the timber too but a mahogany J45 is usually warmer and rounder sounding than a mahogany D18. They also have quite a different response when you dig in with a pick. The Gibson growls while the Martin just gets louder. I'd try to play both and see which you prefer. There may not be much in it when it comes to Epiphones.
Most "mahogany" guitars these days use sapele akaa "african mahogany", a different timber from the classic Honduran type used in vintage guitars.