I don't own a high-end turntable, nor do I have superhuman ears, but I do clean my records and have a decent sound system. I'm sure I'm not getting the most I possibly could out of my music on vinyl, but there is a noticeable difference when I listen to some albums. For example, my vinyl copy of Toto from 1978 has a very full and warm sound, but the digital version I downloaded from iTunes? It sounds tinny as fuck.
I don't buy records because of sound quality, though. I buy them because I enjoy the large artwork, physically owning music, and the ritual of playing it on a turntable.
That's the case where that claim is actually valid. A lot of really classic records made before, I'd say around the '90s(?), sound godawful in a digital format. When I first heard Pink Floyd's The Wall, it was through digital and I couldn't stand it. Even in lossless format, it sounded so cold and digital. The vinyl, however....holy shit. IMO, "the warmer sound" excuse is only valid with old records.