What would cause a 97 Dodge Neon not to start after the airbag was deployed?

Very likely the force of the impact also tripped the roll-over switch. Get your owner's manual and find its location. It will probably be mounted behind a trim panel in the trunk. This switch turns off the fuel pump so the engine will stop running in the event of a roll-over. Re-setting it is a simple matter of pushing a button. Finding it is the hard part.