Projected images are designed to be observed by human eyes. Projectors and screens apply additive color theory to match colors in the real world. When you take a photograph of a real scene. That scene contains the full spectrum of light typically. A screen instead will only contain a set of primaries which are mixed together to create what will appear to be the pure wavelength of those real colors. But only to a human observer.
In real life if you photograph a banana, it's spectral reflection will be real yellow light at around 580nm. Where if you photograph a banana on a screen, the screen will transmit a metameric combination of red (650nm) and green (550nm) light which produces that same 580nm color. But the problem is that the combination only produces a color that appears to match 580nm to a human observer.
Here's a bit of basic color theory.
Spectral power distribution of light source(λ) * Spectral reflectance / transmittance of object(λ) = Color stimulus(λ)
Color stimulus(λ) * CIE 1931 color matching functions(λ)(c) = XYZ tristimulus values
The tristimulus values define the spectral responsivities of the human eye when viewing a particular color stimulus. The color matching functions represent the chromatic response for each cone in the human eye. What's important to note here is that 580nm light and a combination of 650nm and 550nm light will produce the same color to a human observer if the tristimulus values are identical.
Since projectors are designed to metamerically match to a human observer, digital camera sensors on the other hand have their own set of spectral sensitivities determined by the quantum efficiency for each rgb filtered photosite.
Color stimulus(λ) * Spectral sensitivity of camera sensor(λ)(c) = camera spectral responsivities(λ)(c)
Those camera spectral responsivities are different to the spectral responsivities (tristimulus values) of a human observer. Thus the combination of RGB light that was used on the projector to produce white to the eye, does not produce white on camera.