Easier if you asked which is the odd one out
Wingless wet = any wet fly that does not have a solid wing. Problem with this classifcation is that it would suggest that imitative patterns tied without solid wings do not suggest the presence of wings.
Spider = cannot be defined in any words that i can put to-gether.
Soft hackle - fly tied with game or hen hackle , could be a spider, flymph,wingless wet, an emerger, a dun , a bait fish imitation, could be anything.
Flymph = furry nymph , hackled with fur or feather.
Emerger = Emerger
Most anglers I know for the most part used Wets, Dries,Nymphs,Lures as the main classifications. In recent times for some this has expanded to include Emergers. When asked what they are using that is likely to be the response, if you are on really friendly terms they may be more specific
Do we really know what our imitations are taken for, For example the good old hares fur nymph, tie it with very tight dubbing and its predominantly a nymph, incorporate some hackle or a lot of loose dubbing and it could be a nymph, an emerger, a drowned, a still born etc.... How its tied and how and where and when it is fished are inexorably linked.
So is your flymph a nymph, is your wingless wet a lure , a nymph, an emerger , a spinner, a cripple ?
Is your spider a lure, a dun, a spent, a nymph, an emerger ?
Is your spider a flymph, is your flymph a lure, is your wingless wet a spider, is your spider a Wingless Wet, is your lure a soft hackle.
Me thinks these classifications are an incestuous lot
Williams answer is interesting, and is as close as you probably get to what we think these things to mean.
Wingless Wets: the most generic term, but conjures up a classic wet fly...minus the wings. fished swung.
Spiders: slim, sparse, sexy, minimal. two ingredients - silk and game hackle. On face value , yes, but many spiders incorporate fur, peacock thoraxes or touch dubbed bodies
Soft-Hackles: nearly as generic as wingless wets, but at least connotes an attitude in the fishing method and a more complex pattern/use of materials.
Flymphs: associated with Liesenring and Hidy, but defined by the dubbed fur body and attention to overall liveliness of the entire fly.
Emergers: conjures up a method or expectation rather than a fly type. Linked more directly with the entomology and stage of developement which includes opens up an endless types of patterns.