Doesn't change anything gameplay wise other than visuals. I think it would look nice if a certain type of log dropped from certain trees but can still stack with other logs.
Klei has done this in the past with Pig Skin / Pig Skin? and different rock sprites, so I think it would look cool if it was added to birch trees or other trees too (like lunar, spiky, or mushtrees)
Ex:
Birch log drops from birch tree