Neutral in that you lose the flavor of the spirit that was in it and the overpowering vanilla tones down and brings out some of the other flavors. If you’re going for a bourbon-barrel aged beer each time, you’ll probably be disappointed and cubes or spirals would be a better route. But there’s a lot of other flavors you can get in a wood aged beer from a barrel.
Take rum, for example. Rum is often aged in second-use bourbon barrels because the bourbon extracts much of the vanilla from a fresh barrel. Bourbon is aged for only one year, which is why it pretty much tastes like other bourbons, with varying degrees of quality. But take that same barrel and age a different spirit, and you get a completely different character.
Take a cheap aged rum from one of the industrial producers. It’s generally aged for 1-3 years, but still retains some of the vanilla with the industrial solvent flavor. Then a better quality rum aged 5 years, you’ll start to pick up some burnt sugar and caramel. Better rum at 10 years might get some spiciness, and even better stuff at 20 years can have chocolate or orange peel. Most of this comes from the wood and slow aging and oxidation over time. It just takes a long time for the breathing of the barrel to transport some of the sugars deep inside the oak into the spirit.
So it becomes more neutral because the flavors you get from the barrel are changing as some are depleted, but you’re also extracting different sugars the longer you age a beer in it. Can’t rush time.