Which one is more massive; a spinning planet, or an otherwise identical but non-spinning planet?
The simple answer is that the spinning planet is more massive.
The easiest explanation I can provide you without getting into quantum mechanical math, is related to the famous equation E = mc^2. This equation tells us that matter and energy are related, and in a way, matter IS energy, and vice versa. This has been observed at the atomic scale in nuclear fission experiments, where the sum of the product masses of a fission reaction are less massive than the original nucleus, due to a loss of energy. Basically, what we measure as "mass" is really just the effect that an object has on space, time, and other objects around it. It isn't a discrete thing, in the way that you can hold up a penny and clearly identify what is, and is not, that penny. Mass is more like an equation, with several inputs and outputs, some of which are not easy for us to see because of the nature of our perception.
Between any comparable objects, the one that has more energy will also have more mass, i.e. more of an effect on space, time and the other objects around it. For the example of the rotating planet, this effect is called "frame-dragging". To understand this, it's important to see that what is being affected is space-time, i.e. the shape of the universe, and not the object itself. The rotating planet does not inherently have more matter inside it; rather the effect it has on the space around it is different, due to the energy that it has.