Acid and mushrooms are distinct, but they're more similar to each other than either of them is to anything else you're likely to have tried (i.e., anything else except yet another related 5HT2a agonist psychedelic). Seeing pink elephants walking down the street and shit like that is no more likely on LSD than on mushrooms. Usually, the visuals are a sort of secondary part of the experience; the 'mindfuck' is the main course, as it were, and if you've ever tripped on either drug you know how hard it is to put it into words.
Compared to mushrooms, LSD lasts longer, the body high (for me at least) is more of a stimulant/speedy feel rather than the stony trance state that mushrooms seem to bring out of me. My mind races at a million miles a minute, whereas with shrooms it feels more like a dreamy haze. The visuals tend to be more or less similar (breathing walls, color shifts, abstract patterns emerging out of ordinary objects, etc.), but with mushrooms I am more likely to see an actual entity or being (like the shape of a person or animal) whereas LSD visuals tend to focus more for me on abstract patterns and small symbols.
Higher doses of LSD get way crazier than mushrooms ever have for me. Seconds can stretch into minutes, the euphoria can be even better than MDMA at times and visuals will swirl around in pretty much every empty space- I'll look up at the sky and what would normally be just a handful of faintly visible stars through city pollution becomes a tapestry of moving, abstract, colorful shapes and symbols. Overall, I greatly prefer LSD to mushrooms, largely because it's much easier to push into higher doses with LSD.
As for why it's so rare... Basically, it's really hard to make so there are relatively few chemists producing the whole world's supply, law enforcement hunts them down as aggressively as they can and in most countries getting caught manufacturing acid is pretty much guaranteed to earn you several decades of prison time... It's always around, but it's not in ample supply in most areas, and most people who sell it are wary to open up to new customers since it's so illegal and so heavily targeted by undercovers.