Popcorn ceilings do provide acoustic benefits - the tradeoff is of course that most people find them aesthetically objectionable and the level of noise depends on your tolerance.

I just removed my cottage cheese popcorn ceiling and I find the tradeoff tolerable but since I live by myself I don't have issues of sound from other people. I do live in a high rise so I suspect that I might hear my neighbor above me without the acoustic deadening but my ceiling - their floor - is thick cement so the soundproofing is pretty good and if they install hard flooring, they must also put in thick layers of cork.

At the risk of sounding like a broken record, popcorn installed in older homes generally contains asbestos so if you remove it or disturb it, you need to have it done by a certified asbestos remediation company. I had mine removed at the start of my recent remodel by such a company. It protects you and also protects the environment because asbestos is sent to a special hazmat containment site and not sent to standard garbage facilities.