Generally it's against gaming etiquette to pull out a notepad and to start scribbling away during a game. That said, I know some gamers who do it and some games that encourage it, so it seemed like an interesting idea to bat around the notion of paper at the table and think about the impact is has. Some games provide paper and integrate it into the game's design, so there's room to observe and improve upon the use of paper.
Paper as Long Term Memory
The most common use of paper is as a long term memory aid. In this context I mean "long term memory" in the psychological sense of lasting more than 20-30 seconds rather than implying that people write things down because they intend to access them after the game (though in some cases this happens.) Information is recorded with the intention that it be recalled on future turns.
Gamers seem most inclined to use this approach out of place for games where public information becomes hidden but does not change. For instance some players will record Small World scores, as the points are public when scored, then become hidden and critically do not change while unobserved. I think that most players find this sort of behaviour unsporting, but as always I hold to the notion that concepts like that only apply within your local group. Anything goes as long as the people you're playing with agree that it does up front. That being said, I would be inclined to classify this sort of thing as a house rule, since it clearly changes the game in a way that's not in line with the designer's intentions.
Games tend to use it as a means to recording information in a way that does not require a lot of components. I still use pen and paper to track score in my old copy of Titan, it'd be fairly wasteful to produce a big pile of counters to track one variable that can wind up in the hundreds or low thousands, compared to just grabbing a pad and pen. I suspect that the Fantasy Flight (EDIT: I have recalled this incorrectly, Valley Games published the more recent edition. Thanks to Bill Bennett for the correction.) remake uses counters though, they sure do like them
Paper as Secrecy
Paper has the advantage of easy secrecy. It's not hard to hide a piece of paper from someone or to pass it around so that only some people around the table have read it. Cards can achieve the same results, but are less flexible in the messages they can communicate. Dice can be rolled secretly, but keeping the results secret while manipulating other components isn't very viable.
Gamers tend to use paper this way mostly when they desire secret communications with another player. A lot of games aren't explicit about the extent to which players can communicate with each other openly or secretly (and the way in which players react to that could be a whole extra blog post!) Still, some games make it clear that secret communications are encouraged or thematically appropriate.
Games that use paper in this way often use it as a record of secret information, combining it with the memory goal above. Hidden movement games such as Scotland Yard get a lot of use out of a player being able to record their secret information in a way that can be interrogated later. In that particular genre it's worth noting that Fury of Dracula used a card based solution to the same problem that offered significant advantages, such as being able to reveal part of the information without risking showing everything.
Paper as Working Memory
The amount of simultaneous processing that you can manage internally is significant but not limitless. You can do basic arithmetic in your head, but 3 x 5! + 15 / 12.82 will have most people reaching for a calculator despite strictly being a combination of simple operations. A pad of paper can make it possible to work through this sort of problem by doubling as storage for intermediate steps.
In a regular game, players will grab a scrap to perform calculations on whenever they get particularly complicated. The left hand margin of my Pathfinder character sheet is a mess of poorly erased calculations from various points in the game where I'd had more modifiers than sense. This is probably more common in roleplaying games and war games than in board games, but I'm sure there are some stubborn examples of games that refuse to be easily played mentally.
Games use this when they deliberately set up a complicated problem that they expect a player to apply a high level of deductive reasoning to. The first example that came to mind was the difficulty of solving a su doku in your head, but that's more of a puzzle. The most recent game I've played that took this approach was Sleuth, which can best be described as Cluedo without the roll and move bits. No player could reasonably be expected to hold on to all questions and answers mentally and make appropriate deductions, so the game comes with an appropriately designed pad of paper for making notes on.
The games that I cite as providing paper are all quite old. Scotland Yard and Titan are both products of the '80s and Sleuth is from the '70s. There's certainly been a move towards "Games should contain all of the components that are needed to play" in recent years, but that doesn't mean that there's no room for paper. 7 Wonders uses a small pad of paper for scoring (working memory aid) and there are plenty of stealth games that still prefer paper maps for secret movement. It seems that you can get away with it as long as your paper contains some sort of visual aid rather than being blank, even if a blank pad would serve just as well.
There's a difference between what you can get away with and what you should get away with, so the question for a designer is where might paper aid a game? I think that the trick will be to look out for times that players are struggling with long-term memory, working memory or desire to communicate secrets. From that any deliberate struggle needs to be eliminated, some games are hard in some ways for very good reasons (and this is often overlooked by players reaching for paper) and consider alternative solutions. Fixed information (a number, a named location) can often be dealt with differently, but fluid information (a secret alliance proposal, an unnamed location) benefit a lot from this approach. In some ways gaming has grown out of its roots, but in other ways we can benefit by remembering the past and seeing where we can apply it.