Forestry is a well-understood topic that has made wood one of the most widely-used renewable resources in the world. Burning wood still produces large amounts of carbon dioxide/monoxide.
Burning wood for heating is the diesel car story all over again. The tradeoff makes zero sense.
Wood doesen't have to be burned in X place, and it _could_ be gathered and processed using clean procedures, but they're outside of the scope of wood being carbon neutral.
In that case, unless the entire chain uses renewable energy, it can't be considered carbon neutral.
If you burn existing forest and do not replant it it does produce extra CO2.
Also, even if the process as a whole is a net neutral over time, the system requires some amount of CO2 buffering in the atmosphere. The more you use wood as a fuel, the more CO2 you'll have in the atmosphere as part of this buffering.
Or, put differently — "net neutral" just tells you that you can reach a steady state, but doesn't tell you how heavily polluted that steady state is. Higher numbers of wood burners in use still translates into a more heavily polluted steady state.
That's why it is usually considered carbon neutral.
Plus burning stuff isn't that great a way to make power, especially wood - if you have ever tried to heat a house with wood verses a coal fire you'll understand.
Right now the wood pellets are just a way to skirt the rules - I doubt they'll be part of the mix for long, as solar/wind drops in price, along with cheap and reliable storage.