A narrow, but arguable difference. Exploits are ways of using the game's own behavior (usually unintended behavior and glitches) to gain an advantage that the developers did not intend. Speed kissing and getting multiple collectibles through pausing are examples of exploits. (They are an exploitation of glitches to get what was probably not intended that we have - easily gained skill, more tech points, whatever)
The definition has come to include ways of using mods or <cough> inventive programming, to change the properties of the game. As an example, think of challenging a VV player to a collectible hunt, only you're using mods that make the trees, villagers, and statue translucent, and the collectibles and Collectible-sparkles bright, solid colors, easy to find. While it doesn't perfectly fit the definition I posted earlier, such things have come to be defined as exploits in many multiplayer and massively multiplayer games.
The most important distinction though, is that exploits have always been posted publicly here without attention from the ops, including the one I just posted.