Chaos Mode adds endless waves of enemies, as players get perks and increase their combo multiplier by surviving. The update also introduces two new perks: Juiced and Fast Health Regen. The full patch notes are pretty extensive, including mostly fixes to exploits and bugs.
Extinction Mode has been getting regular updates through the season of DLC packs, most recently with the Devastation pack. It hit Xbox systems in April and is due on other platforms in May. Presumably, Chaos Mode will be available through the additional stages still to come through DLC as well.