How does weapon durability work in Fates?

There seems to be a big difference between Fates and the other Fire Emblem games, my weapons don't seem to have a number next to them telling me how many uses I have left.

Staves, however, seem to still have limited usage.

Did they just get rid of weapon durability or does it work differently now?

Correct, weapon durability does not exist in Fates (for any path - Birthright, Conquest or Revelations). Staves however, do have limited number of uses; the stave will go away after it's all used up. In the Birthright and Conquest paths, there is no way to combine Staves into a single item with more uses; staves can be combined in Revelations.