In Revenge of the Sith after Luke and Leia are born, they are sent to separate parts of the galaxy. The droids are given to Captain Antilles, with the instructions to clean them up, and have C-3PO's memory wiped. Why just C-3PO's and not also R2-D2's?
Initially I thought it was due to R2-D2's inability to communicate verbally, but he not only can communicate via text message (as seen in The Empire Strikes Back I believe), but we see other characters understanding his blips and bleeps as well.
This is mentioned in the film's official novelisation.
Echoing @Envite's answer, the fact is that C-3PO opened his fool mouth at the wrong time;
“I’m placing these droids in your care,” the Senator said. “Have them cleaned, polished, and refitted with the best of everything; they will belong to my new daughter.”
“How lovely!” C-3PO exclaimed. “His daughter is the child of Master Anakin and Senator Amidala,” he explained to R2-D2. “I can hardly wait to tell her all about her parents! I’m sure she will be very proud-“
“Oh, and the protocol droid?” Senator Organa said thoughtfully. “Have its mind wiped.” The captain saluted. “Oh,” said C-3PO. “Oh, dear.”
Quite clearly, C-3PO is unable to shut up, while R2-D2 is quite able to keep a secret.
I am unaware of any direct statement on the reason for this by anyone involved in Lucasfilm, but after the release of RotS, the Expanded Universe trilogy The Dark Nest was published, in which Luke finally discovers the identity of his mother due to a malfunction in R2-D2. R2-D2 had kept the knowledge that Luke's mother had died in childbirth from Luke as part of his "owner-protection protocols," which explains why the droid had not revealed Luke's true identity or that of his mother (or father, prior to TESB) during the original trilogy.
It is highly likely that George Lucas had input on this storyline; it was Lucasbooks favourite - and the favourite of no Star Wars fan ever - Troy Denning who wrote The Dark Nest Trilogy, and there is a good chance he was given that role to keep other authors and comic book writers from fighting over who got to reveal Amidala as Luke's mother in the EU. This was after the switch from Bantam Books to in-house publisher Lucasbooks in 1999; George Lucas took a far more hands-on role with the EU at this point, to its detriment.
EDIT: To add to my answer now that I'm home, the novel that contains the revelations about Luke's mother is The Swarm War. There is also a section of this in R2-D2's Wookieepedia entry:
In 35 ABY, when Luke was fixing R2-D2, the Jedi encountered a recording of his father and mother discussing a dream of Anakin's in which his mother had died in childbirth, and another recording of Padmé talking to Obi-Wan Kenobi about Anakin. The recordings, however, were apparently encoded, probably by Alderaanian technicians. R2-D2 himself refused to show Luke or Leia the recordings and insisted he was trying to protect them. The only way to remove the recordings safely was to either find the original designers of the unit, a prototype, or erase R2-D2's memory, an idea only C-3PO seemed to take seriously. Because the recordings hadn't been removed, R2-D2 began to malfunction. By 36 ABY, a year after the first recording was shown, R2-D2 seemed to be functioning again, though he remained unwilling to show the recordings. It wasn't until the Swarm War, when Luke needed to see it to help him in his fight against Lomi Plo, that R2-D2 finally revealed Padmé's death to Luke, as well as his and Leia's birth, and an image from a Hologram inside the Jedi Temple showing Vader and the 501st carrying out Operation: Knightfall at the exact moment Artoo hacked into the Jedi Temple Security system and watched Order 66 happen at the Jedi Temple on Coruscant.
Unfortunately, I still can't find a reason for this, but it would seem it was done so the cheap, cop-out reveal could be executed in The Swarm War.
During all 6 movies we very rarely come across other protocol droids, but many astromechs are on screen. In one scene, R2D2 is repairing a ship in flight mid battle with a few other droids all of which are destroyed. Astromechs were also present in most fighters, of which most were destroyed on screen.
Astromechs are low value grunt workers who are are frequently destroyed on screen, and I would assume would be treated as such by the people they were assigned to.
The flippancy with which Bail Organa gave the order to wipe C3POs memory shows how little he valued the robots individuality. To him, it was a tool, and a tool that had vital information unnecessary to its function. Why bother wiping the Astromechs memory when it shows little communication skill and will probably be destroyed at some point soon.
No doubt Bail Organa instructed R2D2 (and other droids) into innumerable hazardous situations between Episodes 3 and 4 with as much concern as he has for any other tool.
R2-D2 was obviously part of the Rebellion beyond what anyone would believe a droid to be.
Now, with Star Wars Rebels, it is pretty obvious they are setting up R2 as an interegral member to the Alliance/Rebellion, as he has already shown up without any accompanying characters (save Ahsoka), but the funny thing is that R2 could even reveal Darth Vader's identity to her - and he hasn't.
Plus, it's pretty obvious that Star Wars Rebels is trying to apply a more human personality to droids by having the Chopper character.
Oh well, who knows what they will do, now that Disney scrapped the years and years of Extended Universe.
The real answer comes from behind the scenes. Lucas to this day envisions Star Wars as being narrated from a point in the future (hence "a long time ago") and R2 is that narrator. So R2 had to have all memories intact. Unfortunately there is no logical in story answer.
The answer to this question should be fairly obvious to anyone who has watched all the movies.
Star Wars, (later re-titled A New Hope, Episode 4) was the first movie released. At that time they couldn't have any idea that this movie would be the blockbuster that it was spawning so many sequels and prequels. Point being it was episode 4! When it came time to make the prequels, the writers had to reverse engineer some plot points in order to explain why the characters behaved a certain way in what is actually a later timeline.
In episode 4, (the original Star Wars movie), when Luke accidentally plays back a portion of the video file of Princess Leia, C-3 PO is truly clueless as to who this person is an only knows that she was "a person of some importance" on the ship in which they served on. I think it could also be speculated that C-3 PO would be a terrible liar and would not be able to play off his naïveté so genuinely.
R2-D2 is clearly lying through his teeth (figuratively of course), he very well knows the identity of the Princess, has been given a secret mission and tries to hide any information about her from Luke. So it was absolutely imperative when episode 3 finally does get made so many years later that this somehow gets addressed. Wiping C-3 PO's memory or at least a portion of it and leaving R2-D2 and his memory intact fills this plot hole quite reasonably.
