It is a fundamental law of physics that matter/mass/energy (all being, ultimately, the same thing) cannot be created or destroyed — they just change form.
Under 'normal' conditions on Earth, any carbon atom that exists will continue to exist in perpetuity. It can and will form temporary bonds with other atoms to form distinct substances with distinct properties at various points in time, but the actual carbon atom itself will still exist, and the total number of carbon atoms in the world will not change.
(Yes, extraterrestrial carbon sources do exist, but that is beyond the scope of this topic.)
Growing trees obtain all of their carbon from photosynthesis. Carbon in one form (carbon dioxide, a gas) is converted to another form (carbohydrates, which are solids). Those carbohydrates then go on to change form and produce other solids (e.g. lignin and cellulose) that form the bulk of what we know as 'wood'.
During this entire process the number of carbon atoms has not changed. The only thing that has changed is what other atoms have attached to and detached from each carbon atom. No carbon atoms have been created. No carbon atoms have been destroyed.
If wood burns completely and fully in the presence of oxygen, the only thing left behind is ash. Ash is inorganic. Ash does not contain carbon. All of the carbon atoms have — once again — been unbound from and rebound to other atoms. All of those molecules go up the chimney and into the atmosphere. That means all of the carbon atoms are also in the atmosphere. The same carbon atoms. The same atmosphere.
(Yes, a small amount of particulate matter (solid carbon) will become airborne, and some of the carbon will end up in the form of carbon monoxide instead of carbon dioxide, but since both of those actually serve to reduce atmospheric CO₂ levels, we can happily ignore them.)
A tree that 'biodegrades' is nothing more than a tree that is left to rot. Rotting is essentially the process whereby various types of fungi break down the molecules in wood (e.g. lignin and cellulose) into less complex forms. Those forms (and the fungi themselves) are part of a complex biological food web. Rotting wood is 'food' in the process of being 'eaten'.
Unless the tree is completely submerged in water, nearly all of the biological life that is associated with rotting wood relies on aerobic cellular respiration in order to survive. Aerobic respiration is essentially the reverse of photosynthesis (carbohydrates + oxygen » carbon dioxide + water + energy) so the cycle is, once again, complete. The same carbon atoms that entered the tree in gaseous form have been eaten, processed and released in gaseous form by various species of plants and animals.
So, under 'normal' conditions, both processes (burning and rotting)
are carbon neutral. The same amount of carbon atoms go in and out,
and they are in the same form (CO₂).
When 'Carbon Neutral' Does Not Hold
It is under 'abnormal' conditions that imbalances occur — in the short–medium term. An incomplete/interrupted burn will leave behind charcoal that will probably take centuries or millennia to return to gaseous form. A tree felled into a marsh/water will break down in anaerobic conditions and leave behind residues that may become fossil fuels and return to the atmosphere millions of years later.
There are so many variables that you can twist yourself into knots just thinking about them. To actually arrive at an answer you need to reduce the variables — your question needs to be more specific.
Firewood vs Rotting Wood
Wood heaters are specifically built to extract as much heat from firewood as possible. In recent years, with secondary and tertiary burn phases, they have gotten very good at that. Rocket mass heaters (mass heaters in general) are even better. In any case, 'complete combustion' is the goal. Negligible amounts of charcoal should be left behind (or emitted as particulate matter or carbon monoxide). Virtually all of the carbon ends up as CO₂.
In a semi-temperate climate, a tree that is felled and left on the ground, piled up, or placed into a hugelkultur mound, is in a less-controlled environment. It is almost certain that some parts of the tree will decay under anaerobic conditions and thus the embodied carbon will be converted to a more persistent form — one that will remain as a solid or liquid in the soil for (probably) tens to hundreds, but sometimes even thousands to millions of years.
On a time-scale of tens-of-millions of years, there is no difference between the two. On a time-scale of decades, letting the wood decay naturally will sequester a tiny, tiny bit more atmospheric CO₂. The large number of variables makes it hard to nail down "tiny, tiny bit" to anything more specific than "far less than 1%".
Ironically, it has been the 'improvements' in wood heater technology —
the same ones that have made them 'more efficient' and 'less
polluting' — that have made 'burning wood' less useful in
sequestering carbon. Old heaters and open fireplaces
tended to leave behind much larger amounts of charcoal which would
then be incorporated into the garden or spread out into a field/yard —
ultimately becoming buried (sequestering the carbon for hundreds to
thousands of years). Particulate matter (solid carbon) from 'smoky'
chimneys would also settle on the ground outside of homes and
sequester the carbon — in the process increasing the organic fraction
of the soil and improving fertility and crop yields for decades.
Those old heaters and fireplaces were actually carbon-negative!
Wrapping it up...
Efficient wood heaters — thanks to near-perfect combustion — are very good at returning all of the carbon embodied in firewood back to the atmosphere very quickly. They are carbon-neutral and — provided the firewood comes from managed woodlots — completely sustainable.
Allowing wood to decay naturally in a semi-temperate climate will sequester a tiny amount of carbon for decades to centuries. The actual amount is so small and subject to such enormous variability that it makes for a poor and unreliable strategy for the reduction of atmospheric carbon levels over any time-frame.
Many strategies exist that can slow down and/or modify the decay rate of wood and allow it to sequester carbon in the short–medium term, but just leaving it to rot on the ground is not one of them.
Yes. Burning wood is carbon-neutral. Letting it rot on the ground is also carbon-neutral. The (short–medium term) difference
between the two is so small that it should not influence your thinking
in any way. There is no difference between the two in the long term.
If you want to sequester carbon (in the short–medium term) using
fallen trees then you need to actively/artificially manage/control the
decay process — it won't happen to any useful extent otherwise.