The grapple tag info/wiki says:
For questions about grappling — wrestling, or engaging in unarmed, nonlethal combat where the goal is often to incapacitate, restrain, or pin rather than kill one's opponent.
Roleplaying games often have intricate, confusing grapple rules — this tag endeavors to gather rules and rulings concerning this commonplace yet disproportionately complex activity.
The grab tag info/wiki (recently added, after I pointed it out in chat) says:
Grabbing something means taking hold of it so it only moves where you want it to, more or less. The concept appears in many RPGs under various names; grappling, seizing, wrestling, grasping, etc. Most rules for it involve grabbing something capable of moving under its own power and restricting its movement.
Specifically in 4th edition D&D, the grab condition can be applied as the result of an attack, either from the basic grab power or from player or monster powers that represent a grappling fighting style. A grabbed creature cannot move on its own, but can attempt a skill check (athletics vs. fortitude or acrobatics vs. reflex) to escape on its turn, or be moved around as an opposed action by the creature grabbing it, which must also dedicate a hand to maintaining the grab. Grabs can be broken from outside by rendering the grabber unable to take actions, such as by stunning it, or by using forced movement powers to push, pull, slide, or teleport either creature involved in the grab such that the grabber can no longer reach what it was grabbing.
For more general usage involving holding creatures or grappling them, use the grapple tag.
These seem very similar, except that "grab" apparently refers to a different/special thing in D&D 4e (and maybe Pathfinder, judging from the questions tagged with it). Since I'm not familiar with previous editions, I'm not sure if they actually refer to different fundamental concepts (rather than being the same concept with a different name).