There are a lot of generalities tossed about regarding soft tailoring, and perhaps the principal one is that a soft jacket is more comfortable. As a man who owns examples from both styles I submit that soft tailoring has little to do with comfort - a well made structured jacket is as comfortable as its soft tailored counterpart. It is instead about the look and feel of the coat, which on a continuum of stiffness is a close cousin to a cardigan sweater.
That softness is achieved with a tailor-specific combination of hand stitched shoulders, lighter canvas and the use of a piece of wadding in the shoulder instead of a pad. The effect is the opposite of the more military Savile Row cut with its built up chest, nipped in waist and flared skirt, and the difference in structure is principally noticeable when the jacket is draped over an arm. A soft tailoring product hangs like a sweater. A structured jacket does not quite resist gravity's pull and remain horizontal, but it feels as though it would like to.
Compare the soft jacket in the top photo to the middle of the road coat below. The tweed looks as though it might stand on its own if there was no-one wearing it. The flannel looks as though it would collapse.
Both soft and hard tailoring have their adherents with the soft crowd benefiting from the current trend towards informality and the appearance of comfort. But, as the wheel turns, both styles are likely to retain their adherents until the suit passes entirely into history.