We did a physics practical in which I couldn't see the image formed by the convex lens without the screen. But I can see the virtual image formed by the plane mirror without a screen.
You can see real images without a screen. Just look through a converging lens. If what you see is upside-down, then you are seeing a real image.
You are only able to use a screen to show where a real image is because light rays physically converge to a point in space.
When you use a screen, the screen "sees" the real image just as if your eye was where the screen was$^*$. Then we see what is reflected from the screen.
For the plane mirror you are just seeing a reflection, just as if the "mirror image" of your world was placed behind where the mirror if located without the mirror actually being there. Seeing things in a mirror is essentially just like seeing things normally.
$^*$ This is neglecting the fact that your eye is also a lens, so in reality you don't want your eye to be exactly at the image.
This might help. Try focussing the sun with a convex mirror on your hand. You can not merely see it. You can feel it. That is a real image. You cannot catch a virtual image on a screen. You can catch a real image on a screen. (Just as you could do it on your hand.) It is a more convenient way to see a real image. I my school physics lab, we would use pointed needles to determine the distances. We have to adjust the position of the image pin to avoid parallax. When the room was a bit dusty we could actually see the real inverted image of the object pin from the side.