This is a quotation from a semantic book, so I'd like you to not hold this against me. And I think you should know that the book was put out in my mother tongue, so I have to turn it into English on my own. If you come across strange or weird things, please take it into account.
START TO DO SOMETHING and START DOING SOMETHING are generally the same, but sometimes it seems that they have a slightly different meaning each.
'He started to get mean but thought better of it.'
We could assume that he was about to get mean, but he wasn't.
'He started getting mean and I went out.'
He got mean and perhaps behaved badly so 'I' walked out.
BEGIN takes TO DO SOMETHING or DOING SOMETHING as its complement.
In 'Her head bagan to rotate slowly from one side to the other.',
her head moved once; it never did twice or more.
In 'Mr. Swan began making introductions.',
he had some people left to introduce to others, and he'd already introduced a few people.
STOP and QUIT is also close, but for QUIT, the subject, or the actor, must be an animate being. So it would be strange to say, 'The water quit dripping.'
For CEASE, CEASE TO DO SOMETHING and CEASE DOING STH are also similar in general, but they're preferred respectively depending on contexts.
'The baby ceased crying when she heard her parants came in.'
She had been crying, and stopped it when her parents came in.
'That world has ceased to exist.'
Here TO DO SOMETHING would be better, because in this case, the ceasing was not at a particular moment or in a single situation. 'that world' itself was disappeared, or was wiped off and like that. The subject is terminated forever. (This can be proved with the use of Ngram Viewer. The result of searching 'CEASE TO EXIST' and 'CEASE EXISTING' is very noteworthy. The former has been in wide use, but the latter seems never to have been used until now.)
The author says those verbs seem alike at first blush, but each has its own usage, so they should not be sorted in the same category. Also, according to him, verbs like STOP mean finishing something which has been going on, and the form of VERB-ING suggests that the act has started before and still going on, so the verbs and the VERB-ING form fit in well. STOP, GIVE UP types inevitably go with VERB-ING.
This part of the book was written based on works of prominent authors worldwide such as Bolinger, Dixon, Palmer and so forth, so this explanation should be reasonable. (Or, maybe, too theoretical.) I'm not sure I'm getting the content across to you well.
I see that you are a native speaker from British, so it's up to you to accept or reject this quotation. I hope this would be of any help to you.