Barbecue is a lower heat setting, a convectional (heat is transferred through air) heat and is often permeated with wood, charcoal, spice, marinade, rub to augment flavor.
Grilling is an open-flame with a much higher heat setting. Essentially the food is cooked at the grilling surface because well-made grills are designed to localize heat at that optimal place. Grilling quickly seals the outer layer of the food so the surface keeps moisture locked in the food and does not allow smoke, marinade, spiced rubs, etc to be absorbed.
Grilling is often a juicer flavor because the external heat locks moisture in the food. Barbecuing is often associated with moisture because of all the different sauces but most barbecued food is drier and the sauces hide this.
The is no such thing -- in a pure sense -- as a gas barbecue. Gas is incapable of adding flavor and common gas barbecues do not get hot enough to grill. At best most gas cookers are convection ovens, often called broilers.