붙이다(but-i-da, butida)
To stick or attach something firmly to another thing.
When to use
Putting up posters, Attaching labels, Sticking stickers
The nuance
붙이다 is used exclusively for the action of attaching or sticking something onto another object, emphasizing adhesion or close contact.
떼다(tte-da, tteda)
To separate or remove something that is attached or stuck.
When to use
Removing stickers, Taking down posters, Removing labels
The nuance
떼다 describes the action of separating or removing something that is attached or stuck, used only for detaching or taking off.