Aglo I
As far as I know, a cleric in Dungeons & Dragons is almost something like a paladin. Often you'll see them in full chain armour, with a mace or other similar weapon and having holy powers.
However, that is not mandatory. Basically, they're warriors with great devote to gods. Also capable of handling prophecies, healing and such.
While priests, as far as I know, are mostly magic users, and not efficient with weapons.
They're basically the same thing, though Priests are more like your 'In a brown robe tending to the sick, not out fighting for his God, but serving him by issuing Rites, ceremonies etc.'
Where as a Cleric is out there pounding a goblin's brain into goo with that flaming mace in his hand, crying the name of his God while surrounded by corpses of evil doers and covered in their gore and blood. The kind who go out and do the work of their God through whatever means that God wishes to act on force. Take Tyr, patron God of Paladins. He IS a Paladin and demands them to seek out Evil and destroy it.
((I've been playing D&D for almost 15 years ^^))