I started typing this in the Suggestions for the Next RM thread, but realized it was off topic. So I moved it here, but others may have explained this much better.
The primary difference (from my understanding with plug-ins for RM2K/3) is that plug-ins hack into the maker and alter the source code, whereas scripts do not. Plug-ins are more popular with RPG Maker 2000 and 2003 since those programs did not include a script feature. Of course, because of how they are applied, they are considered illegal. And it's been said that if RM2K3 would ever be legalized, the plug-ins could not be used.
Plug-ins are not very flexible compared to scripts. Primarily because, unlike scripts, the user doesn't have to put much effort (if any effort at all) in using them. They simply download the plug-in(s) they want and apply it to the maker as a whole. With scripts, the user typically has to do some sort of configuration, whereas with plug-ins they do not. In that sense, they are a lot like plug and play scripts but without any form of configuration. Unlike scripts, though, a plug-in can access (and even change) the source code, so somebody who knows what they are doing can create a plug-in that can add advanced script-like features without requiring the user to have any program knowledge.
David "Cherry" Trapp's plug-ins are the most commonly used for RM2K3. The David and Goliath patches are probably the two most famous, since it breaks a lot of the limitations within the database, and even allows the user to access the editor even while playing in test mode.
I really don't know why people care so much about plug-ins, since scripts can more or less do the same thing that a plug-in can. For old makers that don't include a script feature, it makes sense. For newer programs, not so much. One reason might be less compatibility issues with plug-ins than with scripts. Another reason might because newbies who are frightened of touching anything script related might think it's easier, and less of a risk, to just install a plug.