The a priori dismissal of the multi-plane idea might be stupid, but the thread is nevertheless good. Leaving out obvious fighting style merits, switching planes adds a great deal of depth. It's like classic hollywood pictures and their efforts to emphasize the immersion through a more three-dimensional image composition. Lighting, depth of focus and so on. Even the studio logos were perspectival, like the ones of 20th Century or Warner.
SNK had always a knack for great backgrounds, which, in contrast to Capcom, were always emphasizing depth. The Korea-Team in KoF '96 shows it best. There's nothing there, but an absorbing vivid feel of width on the z-axis. The different planes in SNK games are just an enhancement of their background concepts: they made the depth playable, physically attainable for the player.
I'm thankful, that SNK was able to use that idea outside of the versus fighter universe, like in the above mentioned Top Hunter. I would've loved to see other genres applying this concept. Just imagine a two plane Blazing Star.
All in all, the gameplay benefit may be minor, but the structural gain is a clear surplus value.