It is easy to treat oak as a flavouring and stop there. The oxygen and the evaporation are doing at least as much, and they are the reason a barrel is not interchangeable with adding wood to a tank.
What the wood contributes
Vanillin, read as vanilla; oak lactones, read as coconut and sweet wood; and from the toasting of the barrel's interior, notes of smoke, spice and caramel. A new barrel contributes far more than a used one, and the effect falls away with each successive fill until a barrel is essentially a neutral vessel.
French and American oak
American oak, Quercus alba, is typically higher in lactones and reads sweeter and more overtly coconut-like. French oak, chiefly Quercus robur and Quercus petraea, is usually tighter-grained and subtler. Both statements are generalisations across a wide range of forests, seasoning regimes and coopers.
Why size matters
A smaller barrel has more wood surface per litre of wine, so it acts faster and more strongly. A 225-litre Bordeaux barrique is an assertive vessel; a large foudre holding thousands of litres is close to neutral and is chosen for the slow oxygen exchange rather than for flavour.