Why does a heated body emit a continuous spectrum of waves (as I understand it, that's why they burn red/white/blue), and a burning body is one color?
as I understood it, for example, the green flame emits certain spectral waves. Why does a highly heated body not emit only one wave, but many several, because the internal energy in the body is the same?
A warm body emits radiation at all wavelengths, according to Planck's law. The wavelength with maximum radiance is $$ \lambda_{max} \approx \dfrac{2,898\times 10^{-3} m\cdot K}{T}, $$ a result known as Wien's law. This the colour you "see the most". From the equation, you get that the hotter the body, the more energetic (bluer) the light will be. Note that the colour you see the most is not the only colour emitted.
Let's use an example: Fire is usually red, but not always. As you get closer to the burning body, fire turn first yellow, then green, and finally blue. This is because the closer you are to the burning object, the higher the temperature is, so the lower the wavelength of the emitted light is!
In any case, from the point of view of radiation, we do not differentiate about burning and very hot bodies. For example, if you heat steel (see videos of sword making), you can see that the metal changes colour as it is heated.
The 'single color' of a hot gas is a spectral emission line. Those spectrum lines are emitted by a single atom, and are perturbed (broadened) if another atom is nearby. That's called 'pressure broadening'.
Solids are always 'pressure broadened' because many atoms are nearby... so an emission rule that ignores individual emission lines is appropriate (the black-body limit dominates). Hot gas of a flame is more colorful because the quantum state of many similar gas molecules makes an important modulation, and we detect that against a relatively small contribution from pressure broadening.
