Recently, both Boeing and Airbus launched "new" aircraft as a modernized version of their respective best sellers (B737MAX and A320neo). To narrow the question down, I will consider the A320 and the FAA, but the same goes for the B737 and almost certainly many more aircraft families, and I hope for other certification authorities as well. I'm not really clear on the difference between type, model and series but I think the difference may be relevant for the certification process. Perhaps highlighting this difference is enough to answer the question.
Wikipedia lists multiple variants of the A318. Do the A318-111 and the A318-121 require different certifications?
Moreover, when adding a variant to the family (here adding the A318 to the family composed of the A319, the A320, and the A321), do the certification authorities consider this aircraft as a totally new one or only as a variant of existing one (many shared components)? Can this consideration lighten the certification process?
Is the new version (A320neo) considered as a new aircraft by the certification authority and does it thus require the same set of tests as a new aircraft?
In short, when a new aircraft is an evolution or a variant of a previous one, how is it considered by certification authorities?