New versus Neu(tral)
What is the difference between new and neutral oak? The short answer is: extractable flavor.
After wine is placed in a barrel for the first time, it absorbs a huge amount of the flavor and aroma compounds that were generated during the barrel's toasting-- vanillin (vanilla), syringaldehyde (spicy, smoky), other aromatic aldehydes (resinous), lactones (oaky, coconut), and aromatic phenolics (clove, smoky). As these are extracted into the wine, the concentration available for further extraction is diminished. It varies from winery to winery, but most folks consider a barrel to be effectively neutral somewhere between the 4th and 6th fill. Neutrality in this case means that no discernable oak compounds are extracted into the wine held in barrel.
This doesn't mean that the barrel itself is useless, though. Despite not contributing to the wine's aromatic or flavor profile via extractables, the oak provides a semi-permeable vessel that can allow a wine the space it needs to soften through slow oxidation, and allows the polymerization reactions necessary for tannins to smooth out to take place.
Not everyone loves the flavors of big oak, and a mix of new and neutral oak can help a winemaker strike the correct balance between the oak complexity and allowing the fruit's varietal flavors to shine.