In the style of the numerous D&D edition comparison questions, what are the major changes that Numenera 2 offers over the original release?
Numenera 2 has two corebooks. Discovery, and Destiny.
Numenera 2 works now in the Cypher System. The main change it brings to Numenera is related to armour. The costs in might and speed associated with wearing armour is replaced with an increase to how much Speed needs to be spent to use Speed Effort when wearing armour. Light takes +1, Medium takes +2, Heavy takes +3. Edge works as normal, and not being practiced in wearing that type of armour increases the effort cost by an additional +1.
In Numenera 2, the distinction between anoetic and occultic cyphers is removed. If you use cyphers from Numenera 1 in Numenera 2, each counts as 1 cypher.
At tier 3, and again at tier 6, a character can choose between several abilities for his focus.
Discovery is similar to the old corebook.
Discovery introduces the concept of Player Intrusion: a player paying 1 XP to do something cool. What a player can do with an intrusion is listed in its type (so a glaive has a different set of player intrusions than a nano). Usually, a player can do this no more than once per session.
Destiny introduces new character types: arkus (leadership), wright (crafting), delve (salvaging). There is also rules for large groups of creatures (hordes and communities), salvaging (introducing iotum, numenera crafting components), and crafting (introducing plans for building commonplace and numenera objects or structures).
New descriptors and foci related to the new types or interacting with the new rules are in this book.
Numenera 2 is compatible with previous Numenera products, including the bestiaries and the adventures, with the minor changes to the armour and cypher rules described above. However, the publisher (Monte Cook Games) does not recommend using Numenera Character Options and Numenera Character Options 2 with Numenera 2.
Numenera Discovery not a new version but an update to some of the core rules. This does mean that all content is supposed to be backward compatible except for those small changes that have been made (list in this other answer).
My group and I have had no problem using the new core rule book with an adventure which was started with characters created from the original book.
If you're a big Numenera fan then you are going to want to buy the new core book, but you don't have too.