One of the differences is that decomposing happens in the form of Methane, which is a potent greenhouse gas (21x as potent as CO2).

Capturing and burning the methane "downgrades" the potency of the emissions, now only emitting CO2.

Also not my area of expertise, but I think the idea behind the biomass == carbon neutral is that the carbon in those fuels was already in the atmosphere a few year ago, so you are not really tapping into the fosil fuels stored in the Earth's manttle. You can consider this a form of low-tech solar, since the biomass had to ultimatelly use photosyntesis to accumulate energy.

Also, if done sustainably, you have to keep sustaining life forms that keep doing the photosyntesis trick for you every year. i.e. All those carbon atoms in this year's crop of Christmas Trees were in the atmosphere a couple of years ago, and we are potentially breathing out at least part of the crops from the next few years... But if someone goes and cuts down a big chunk of the Amazonas in order to make "bio-fuels", yeah, that's dishonest, stupid and self defeating.