The primary difference between Nidoking and Nidoqueen is that they come from different gender lines of the Nidoran. On the surface, the two Pokémon are almost the same with the exception of basic numbers. They possess very similar move sets, they are both Poison/Ground Pokémon and they come from similar evolutionary lines.
This gender disparity is the first and most prominent example witnessed in the Pokémon series. Gender as a full game mechanic was first introduced in the second generation of games, but the Nidorans displayed a basic example of what it would eventually become in later generations. Even in the latest versions, the Nidoran lines display more differences between their genders than others, as they are one of very few to develop different strategies and strengths solely depending on their gender.
The Nidoran lines are considered separate species from each other, but they possess perfect compatibility with each other. Unfortunately this does not carry all the way through their evolutionary lines due to quirks with how Pokémon breeding functions early in the series. Strangely enough, once a Nidoran female evolves to a Nidorina, she loses all egg compatibility, making it impossible to breed with anything other than a Ditto.