They are different rounds. The 44-40 meant .44 caliber and 40 grains of black powder. New loads in the 44-40 use modern powder but that is just where it came from. The .44 mag is a little shorter and not a rimed cartridge. They are not compatible with each other.
No they are not the same length. The .44 mag case length is 1.285 and the 44-40 is 1.305. By a rimmed cartridge I mean that the 44-40 does not have a groove after the rim. It is straight walled where as the 44 mag has a cut groove for the extractor. They are not innerchangable!