Aerobic composting - oxygen, open-air, compost
Aerobic composting requires oxygen to decompose organic waste via naturally occurring micro-organisms. This process produces heat, water, carbon dioxide and small amounts of methane and nitrous oxide.
Aerobic composting uses grid electricity to power the aeration of decomposing organic waste. Composting is great, but it still causes greenhouse gas emissions. Aerobic composting, while far better than landfilling, does consume transport fuel and grid electrical energy.
Dry anaerobic digestion - no oxygen, contained, compost + renewable energy, closed loop carbon cycle
Dry anaerobic digestion decomposes organic waste in oxygen-free conditions. The organic waste produces carbon dioxide and bio-methane.
The bio-methane can be used to generate renewable energy for storage or redistribution, ie. a closed-loop carbon cycle. There are very little fugitive emissions since the entire process is contained in enclosed structures and gases are collected for appropriate handling.
In summary
Both aerobic composting and dry anaerobic digestion produce valuable compost products and both produce approximately the same amount of physical carbon dioxide gas emissions. However, dry anaerobic digestion also produces a renewable energy source and as a result, has a lower carbon footprint overall.