With full demutualization, the original mutual company essentially buys all the stocks at issue once it's fully converted into a publicly traded company. Consumer owners are compensated using these shares, cash, or statement credits, depending on what they choose.
In sponsored demutualization, the mutual company's stocks are instead purchased by a new parent corporation, which then holds all the stocks and ownership of the former mutual company. Consumer owners are then issued stocks for the new company or given cash compensation.
The main difference between these two types is who owns the company once it's gone public. Is it the original mutual company, which may include some older corporate structures that could be difficult to translate, or is it a wholly new company with a presumably modern structure?