There's 2 kinds of randomness I want to talk about:
Dragon breathes fire on Sneaky McThiefenhower, Sneaky rolls saving throw dice, everybody watches, we wait for the result, hey it's random. We all know this situation very well. It is also widely agreed to be a lot of fun. What disease did you get? (roll roll) Haha, gout! You tool! It's fun, its funny, it's a good chunk of what makes the game cool and sometimes exciting. Almost everything's more fun when everybody knows it's random.
Random To The DM But Not To The Players (Secret Randomness)
Player: "If the wind is favorable, we should be able to make North Island by nightfall. Which way is the wind blowing?"
DM:(looks down at the random wind direction that s/he rolled up before the game started.)
Player: "Awesome. You're such a nice DM today, what got into you?"
In this case--and this is the case I want to talk about most here--the result was determined randomly but there is no way the players could know that.
Here's an example from the AD&D DMG (pg 47):
If the PCs are hexcrawling in a "moderate to sparsely inhabited" area the base chance of an encounter is 1 in 12, whereas in an 'uninhabited/wilderness" area it's 1 in 10.
While I recognize and appreciate the Gygaxian Naturalism implied here, there is absolutely no chance that the players ever will (unless I actually tell them these odds or they see me rolling random encounters at the table--in which case it ceases to be Secret Randomness.)
To the players, they either run into a monster or they don't. Maybe if they go through several hundred hexes they'll realize what the odds are like, but if the DM throws in just a few pre-planned encounters then there goes that. It all just seems like it's in the DMs lap. These 1-in-12 vs. 1-in-10 figures are not statistics the PCs can utilize tactically or consider when making their plans.
To a player, this not like the math implied by their Str 6---which tells them that if called upon to perform a feat of strength, they will probably fail, though not as often as they fail to perform a feat of dexterity since their Dex is 3.
In other words, very often Random to the DM But Not The Players has almost no effect on the game other than allowing the DM to roll some dice rather than pick something interesting.
Reasons To Use Secret Randomness:
Before the game, secret randomness sometimes speeds things up. There are ways to do this "wrong"--rolling wind direction randomly and secretly before the game seems pointless. A bad random generator does little more than have rolling for a simple option (left or right?) replace thinking about a simple option.
On the other hand, some tables are so long (like the DMG magic item table) that rolling is faster than reading every single entry. So there's a good reason to roll. A good random generator (like I hope this one is) actually makes several choices at once very quickly. The Another Brick In The Wall generator and most computer generators give you monster placement, room placement, item placement, and paths from room to room in less than a minute. Saving time is a decent reason to do a lot of things, but the trade-off, obviously, is that you think about things less, which is often a bad thing. But you probably know all about this and have already thought about it. Secret randomness during the game is a much more slippery subject...
This is probably the best reason to use Secret Randomness during a game. Concealing the fact that the nearest necromancer lives just next door just coincidentally or that the pub you walked into just happens to contain a guy who has a bizarre proposition concerning waterfowl may make the PCs suspect there's a plot going on against them when actually there isn't.
If the PCs know that the encounter with the (random roll) frightened (random roll) child babbling about (random roll) lava (random roll) trolls was random, they won't feel the need to investigate, but if there's a chance that it was a planned event the DM sprung on them then maybe they ignore the curse of the lava trolls at their peril...
A related sub-reason might generally be because, if the players realize the DM is generating fixed attributes of the world randomly during the game (like street layout) it might throw them out of the "fantasy" for a second. I tend to not think this is a big deal (especially if the paranoia effect is not also in play), but some people do.
However, it's important to note that Secret Randomness for the purpose of generating paranoia is no different, from the players point of view, from Secret Arbitrarily Making Stuff Up. So why use a chart at all?
To Decrease The Number of Things The DM Has To Think About During The Game
Assuming that we're using Secret Randomness during the game because we want to create paranoia or keep everything verismilitudinous, why are we using a chart instead of making it all up? If we use a complicated chart with complex options--the only kind that's worth using before the game--then that usually takes too long either to consult or to read the result during the game (especially considering that you're trying to keep it all secret).
Ok, in my Urbancrawl Rules the shapes of correct paths to PC destinations are Secretly Random. Why'd I do that? Why not just arbitrarily put a place on an arbitrary map and go? Because I know all the paths available in the random generation method I designed are of approximately equal length and complexity so I can rely on it and not have to second-guess myself. However, since it's all based on how a die looks when it comes up, you don't have to actually spend time consulting a chart. Paradoxically, it also might, over time, generate a level of street-plan complexity that might not happen if I was thinking it all up on the fly since we have a tendency to think in repeating patterns. (Maybe.)
I admit, this is probably the worst reason to use Secret Randomness. In order to be both fast and secretive, you have to use a chart with less complicated results than the really involved ones you'd use before the game, but at least as complicated as what you'd be able to think up yourself on the spot, which is a pretty tough niche to fill.
To Entertain The DM
In the urbancrawl rules I recommend writing up four kinds of pubs before each city-based game. When the PCs enter a bar, secretly roll a d4 to determine which one. Why bother rolling? Why not just go through the pubs in order? The PCs'll never know. Same goes for random encounters--why not just put them in order and have the first one be the first one and the next one be the next one, etc.? Because it's more of a surprise for the DM that way.
All together, I feel like the case for Secret Randomness During The Game is fairly thin. In most cases, I think Secret Arbitrariness or Open Randomness is usually going to result in more fun. But you never know.