Covert operations, clandestine operations, and stealthy operations are often confused. According to the Department of Defense, a covert operation is "so planned and executed as to conceal the identity of or permit plausible denial by the sponsor. A covert operation differs from a clandestine operation in that emphasis is placed on concealment of identity of sponsor rather than on concealment of the operation."
As such, it may (or may not, depending on the circumstances) be permissible for the occurrence and details of a covert operation to become known, but it is never permissible for the sponsoring party to become known. This ties in to plausible deniability: a covert operation must almost always be deniable, while a clandestine operation need not always be deniable: if it has come to the point where anybody has to deny anything, there is a good chance that there is no way the sponsoring party has plausible deniability.
Tsarren's writeup makes another common confusion between being covert and being stealthy, the latter being focused on not being detected in order to complete the direct action successfully and not necessarily on preventing discovery of the action or the sponsors after the direct action has been completed. The distinction between being stealthy and being clandestine is not quite so clear, although in general usage while a stealthy operation is only intended to be concealed until its completion, a clandestine operation is often not intended to be discovered at any point in time in the foreseeable future, and possibly ever.
To make all this terminology clearer for the layman, consider the following situation:
A terrorist cell steals a small amount of weapons-grade plutonium from a government facility and hides out in a small, abandoned cabin in the middle of the woods.
- Stealthy but not covert or clandestine: A SWAT team sneaks up to the cabin in the middle of the night, tosses in flashbangs and shoots the baddies. The team's public relations officer, the local mayor, the governor, and the President all make statements to the press declaring the terrorists dead with no friendly casualties and the plutonium en route back to its secure storage location, thanks to the work of the brave men of SWAT.
- Stealthy and covert but not clandestine: A team of men clad in black from head to foot and wearing balaclavas sneaks up to the cabin and shoots all the terrorists. A HAZMAT team comes by to pick up the plutonium, after which the team of shooters load into unmarked, unplated SUVs and drive away before the media shows up. A public relations officer from the White House declares that the situation has been "dealt with" and refuses to comment further.
- Stealthy, covert and clandestine: Some muffled bangs and pops are heard in the woods. When the neighbors go to investigate, they are stopped at the perimeter of the property by a line of yellow danger tape and a slightly nerdy middle-aged man with a clipboard identifying himself as the regional gas inspector. The inspector tells them that it's unsafe to step on the property because a leaking gas line has been causing small explosions in the house. When the neighbors stop by a few days later, the tape and man are gone, and the house is a bit cleaner than before. The plutonium is later rediscovered in a different room of the facility, having been misplaced due to a labeling error.
Some black operations are chosen to be covert rather than clandestine because their effects are too visible to be clandestine, and because it is more difficult, takes more time, and costs more money to keep an operation clandestine.
Also, although I have used a direct action operation example in this writeup, covert operations are by no means all run-and-gun missions. In fact, I would speculate that a majority of such operations are not: COINTELPRO and the events that sparked the Iran-Contra Scandal are both high-profile examples of (formerly) covert operations.
The U.S. Department of Defense Dictionary of Military and Associated Terms - Joint Publication JP1-02, 05 January 2007