What's the difference between "nerf" and "debuff"?

Solution 1:

A nerf is typically a balance change made by developers of the game to reduce the effectiveness of an item or strategy:

January patch notes: Reduced the damage of the diamond axe from 9 to 7

A debuff on the other hand is typically a temporary status effect that is inflicted on players/characters in the game:

I got hit by a splash potion of weakness; my diamond axe only does 5 damage for the next few minutes.

Bear in mind that both of these terms share the same antonym of 'buff', which can easily lead to confusion. This would be a buff (opposite of nerf):

February patch notes: Increased the damage of the diamond axe from 9 to 10

This would also be a buff (the opposite of debuff):

I drank a potion of strength; my diamond axe now does 12 damage for the next few minutes.

Solution 2:

Typically, I would consider a nerf a lowering (or otherwise worsening) of the stats of a race/class/character/boss/playstyle/(etc...) in the game code (often as the result of a patch).

Debuffs tend to be more transitory - you often apply debuffs in-game to lower the stats of something for either a limited time, or until the target of the debuff dies.

There is some room for overlap...