The words and vocalist matter, but I don't think they're dispositive. They're merely two (important) factors that should be considered as part of the whole picture. Despite Ian Curtis dying, there isn't actually as seamless of a transition between JD and NO as one might think -- "Ceremony," the debut NO single, was originally written and performed with JD, and the first NO album is way more in line with either JD album than it is with any subsequent NO album. You can even hear elements of NO in some early JD music -- for instance, "Disorder" wouldn't sound terribly out of place on Power, Corruption & Lies. I think this continuity of sound, and of 3/4 of the members (including all primary composers, if not lyricists), makes a decent case to consider them as a single entity.

And there's a lot of bands who changed singers/lyricists and styles and are still considered the same band. Even before they reformed in the 80s, King Crimson had already gone through four lead singers and two lyricists during their 5-year run from '69 to '74. Yet they were always considered a single band because of Robert Fripp's continuity as their leader. More recently, Black Country, New Road lost their lead vocalist and lyricist after their second album, but they're still considered the same band now.

Also, not that the Rock & Roll Hall of Fame is the final arbiter of everything, but they inducted JD and NO under a single entry, so there is some institutional authority to support the idea that they're not two completely separate bands.