When Eobard Thawne ceased to exist, it should have reset the timeline so that Barry's mother was never killed, Barry grew up with his parents, and Harrison Wells' particle accelerator didn't come online until 2020 (resulting in the creation of The Flash).
However, that didn't happen. Instead a paradox ensued, the result of which created a singularity above Central City. Barry and Firestorm were able to close the singularity, but it seems to have created rifts into at least one alternate reality, Earth-2.
Harrison Wells warned Barry that when you go back in time and change things, the universe has a way of working itself out in unexpected ways. Eddie killing himself would have been a complete rewrite of a lot of history. So instead, the universe decided it could solve all that nonsense by creating a black hole instead.
Despite the supposed never existence of Eobard Thawne, Barry and his Super Friends retain full knowledge of his life and actions.