'Exist' has a connotation of permanence, whereas 'cease' is something which can vary. Something can cease, and then start again.
There are some cases where 'It never exists' might work, but it is sort of clumsy. You might use it in a sentence such as 'This is a specific object. It never exists without a certain attribute.' I'm actually struggling to think of another verb which can't follow 'never' in a present tense. I think 'exist' is the special word here, rather than 'cease'.
'It never ceases' implies that it is something that often has the chance to stop (and possibly start again), but never does. Equally, 'My car never starts when I need it to' or 'It never rains in this town' work fine.