A damage meter typically does what it says on the tin, it shows you your current dps. Some might show you a history of your performance, but the key is that a dps meter is really about YOU not someone else.
A Rift combat parser, on the other hand is generally more full featured. It is intended to show the full picture of combat. You generally get a way to select the particular information you are looking for. So you can slice healing per second away from damage per second and examine the performance of your healers. You can go on to see who died and why they died.
A good example of a full featured parser is worldoflogs.com for Word of Warcraft. Another example might be the WoW UI Addon called Recount. Where Recount touts itself as a dps meter, it is really more of a combat parser.
Examples of Rift Combat Parsers
2012/01/08 Edit: An anonymous commenter pointed me to a DPS meter called "Simple Meter" available on Curse.com. You can take a look at it and decide if it's something useful for you.