Soft v Dry
Soft hackles are "usually" Partridge, Pheasant, Starling and similar feathers that are supple and impart motion when fished subsurface. There are some types of rooster feathers that are considered soft hackles, but most "soft hackle" flies use non-rooster feathers (but there are always exceptions).
Hard hackle as you called it, or dry fly hackle comes from rooster capes and saddles. A good dry fly hackle has lots of uniform barbs packed into a small area that flare out very distinctively and are strong enough to support the weight of the fly on their own. The barbs on a Dry fly are usually 1x to 1.5x the hook gap.
A soft hackle will not support the weight of the fly and the barbs are usually tied to be just less than or slightly more than the hook shank.
There are all sorts of exceptions, but those are the general principles.
"Even a fish wouldn't get into trouble if he kept his mouth shut."