Is 4K worth it over Full HD in a projector?

At the sizes projectors exist for, yes, 4K is visibly worth it. A big picture is the entire reason you chose a projector, and Full HD is the resolution that struggles most as the image grows.

The reasoning is simple. Resolution is fixed; screen size is not. Spread 1080 lines across a 100 or 120 inch image at a normal seating distance and you see the softness and, on some models, the pixel structure itself, especially on fine text and diagonal edges. Push the same source through a 4K projector and the picture holds together at distances where Full HD has started to look like a very large television from 2012. Is it measurable? Yes. Is it audible? No, but you knew that; it is very much visible.

There is a second, less advertised benefit. Current-generation 4K models carry the current generation of processing and HDR handling. Full HD projectors are largely older platforms, so you also give up tone mapping and input features that have nothing to do with pixel count.

Now the nuance the box will not explain. "4K" is implemented differently across projector technologies. Some use native 4K panels; many use pixel-shifting, where a lower-resolution panel is shifted very quickly to put more distinct pixels on the screen. Both are sold as 4K, both look sharper than Full HD, and they are not identical. The manufacturer's spec page tells you which method a given model uses; I would read that line before I read the marketing paragraph above it.

Where does Full HD survive? Budget rooms, casual use, smaller images, and situations where the money is better spent on light control or a decent screen. A well-set-up Full HD projector on a proper screen in a dark room still beats a 4K projector fighting a sunlit wall.

For a 100-inch-plus image with 4K sources, 4K is the sensible floor today. Tell us the room, the size and what you watch, and we will say which implementation makes sense at your budget rather than which one has the biggest number on the carton.