We know that a plane mirror can reflect rays more precisely so we can get a more accurate image. So why can't we use a plane mirror instead of a projection screen?
Using a plane mirror would be a problem only because it can reflect the rays more precisely or what can be called as a specular reflection.Using a screen rather than a mirror would make the reflection diffuse in almost all directions equally called as a diffused reflection.Using diffused reflection rather than specular reflection is important and this could be explained using a simple analogy.Consider the objects you see in daily life they can be seen from any direction because they diffusely reflect the light whereas the image you see on a mirror changes with the direction of your view.Using a mirror would make the image appear different for different audience.The left part of the diverging beam falling onto the mirror would mostly reflect to the left side and can only be seen by the audience on the left side and vice-versa.This wouldn't be a case with a screen which diffuses light falling onto it.
No Problem, you will end up getting a much clearer image provided you can clearly see faraway things. The reason I said that is because by using a plane mirror, the image will be formed behind the mirror. The reflected rays will not reach you directly, but further form a virtual image of the projector's image.