What are the advantages and disadvantages of the hydro-pneumatic controls against the hydro-mechanical controls?
They are actually the same thing. Hydro mechanical fuel controls often include the use of pneumatic pressure to regulate the fuel flow. Here’s an example of a simple one: http://blog.covingtonaircraft.com/wp-content/uploads/2015/08/fuel-control-unit-pt6a.gif
The mechanical part is the input of rpm by a shaft connected to the compressor spool. The pneumatic part is the P3 (compressor exit pressure) input.