Like Mr.E said, the both Nidos are virtually identical. (Revelant things on bold)
Nidoking had 7,1% more of Physical Attack and 7,4% more of Special Attack.
Nidoking speed tie with; Aipom, Articuno, Girafarig, Gligar, Golduck, Heracross, Kingdra, Ledian, Misdreavus, another Nidoking, Pinsir, Qwilfish, Stantler, and Suicune, and outspeed; Gyarados, Arbok, Dragonite, Kabutops, Meganium, Venusaur, Blastoise, Feraligatr, and doesn't speed tie with a Nidoqueen or Hitmonchan.
By other hand, Nidoqueen had 7,4% more of Physical and Special defense + 4% of HP. Nidoqueen heals 1 point more of HP with Leafovers per turn (both are really near to heal 1+), and Nidoqueen can't create 101 HP substitutes or something like that.
About it's movesets, Nidoking had; Thrash, Horn Attack, Confusion, Fury Attack, Amnesia, Disable, Leer, Morning Sun, while Nidoqueen had; Bite, Scratch, Tackle, Double Kick, Fury Swipes, Charm, Growl, Moonlight, Tail Whip.
That's is all the difference, Nidoqueen actually had the incredible and interesting move Charm, so, it allows her to check Curse users and counter the ones without Earthquake and force switches, while Nidoking can use Disable if someone really wants, the recover moves are not so much used, and Amnesia Nidoking is not really effective at all. Maybe the unique revelant thing is the Nidoking's speed, as it leave him able to at least speed tie more threats, and outspeed some important Pokémon who could otherwise make him too much damage or take advantage of a free or easy switch, but that's pretty much all.
Also, the possible discussion is if Nidoqueen should be BL or not, and if down a BL or two could be a good idea or not.