Let's say you have a roleplaying game with these elements:
How is this not a board game?
(The question may sound antagonistic, but I'm quite serious. What is it that makes such a game not a board game?)
To give a short, simple attempt at answer to kick things off: I would say that a roleplaying game must be partly comprised of hidden information that has been invented by one or more of the players.
That's the key difference I can see between The Arkham Horror/Mansions of Madness (boardgames) and The Call of Cthulhu (roleplaying game), where both are moderated by a Keeper. In both setups the Keeper controls the opposition to the players, but in only one is (s)he asked to invent details, descriptions or outcomes, as opposed to just managing them.
In a boardgame, it would be seriously frowned upon for any player (even in a moderatorial role) to come up with new challenges or house rules for the others that are not in the script. In a roleplaying game, that's practically compulsory.
In an archetypical board game, the action focuses almost exclusively on that which can be represented by state on the board, state on a sub-board, or marker substitutions. Further, board games generally operate on the principle of "What is not explicitly permitted is forbidden."
In a Roleplaying game, the board is not the final arbiter of what is present - it's the "Gamemaster." Further, a great many RPG's do not use a map on the table at all in play; while the minis-on-map crowd has been with us from the dawn of RPG's (I was told this by Dave Arneson via email in 2005 or so, and have seen it since 1980), it wasn't the "normative method of play".
Roleplaying games, in general, operate as a radioplay - people talking about their character's actions and speaking their character's dialogue. Even when the board is present and used, it's secondary to the actions as described.
Board games tend to have fairly precise, albeit often abstracted, rules. The general principle in board gaming is you can only choose actions for which the rules allow.
Roleplaying games, in general, take a different approach - what isn't explicitly forbidden is able to be attempted, and rules only delineate what's assured and what's forbidden, leaving a whole gray zone of "This can be attempted but it's up to the GM to figure out how to resolve it." Some, going back to 1975's T&T, strongly encourage "making stuff up on the fly." Including house rules to cover issues specific to your group and play style. To a lesser extent, even the 1974 edition of D&D encouraged house rules.
There are several commonly held beliefs about RPG's that are fallacious. I'll address several of them in the spirit of completeness.
Hidden information is neither requisite for RPG's, but strongly discouraged in a few (like Houses of the Blooded and Burning Wheel).
Likewise, it's not alien to board games. Stratego is a simple form of hidden information constructed by a player and kept from the others. More detailed forms include any board games with double blind play.
While it's normative to have a singular Gamemaster running RPG's, several do not require one, usually sharing the duties amongst the group.
Likewise, most double-blind games require a referee, and many wargames of the 1960's and 70's presume a 3rd player will serve as referee. Not a few include the provision in the campaign rules for the GM to introduce surprises of their own.
Both of these are present in a number of board games. FGU's Star Explorer has continuing campaigns of multiple plays, and character advancement of the captain. Car Wars by SJG, Star Warriors by West End and BattleTech by FASA/WhizKids likewise allow driver/pilot experience to improve in continuing non-roleplaying campaigns.
While they are normative to RPG's, not all RPGs include them; a few are intended for one-shot play per character, and provide no means of character growth.
It's required in Aye, Dark Master, a card game.
It's not actually required in most RPG's, the rules working just fine with the abstracted "My dude's gonna try and charm the society lady using my savoir-faire skill." It's not the norm to play in the abstracted third person, but it works.
No, there isn't.
Car Wars has been straddling the line since the mid 80's... neither fully board game, in that there are suggestions on how to roleplay in Deluxe and later rulebooks, but also not fully RPG as the game is played strongly tied to the map-state.
Battlestations likewise sits just a hair on the boardgame side, but the authors themselves refuse to class it as purely either one.
Burning Empires is often described as a narrativist wargame in roleplay mode.
Diplomacy is as much a freeform roleplaying of diplomats as it is a wargame. Several of its derivatives are likewise strong on the off-map action being as important as the on-map/on-component tracking.
How is a board game not a role-playing game? Let's say you have a board game game with these elements:
an Overlord (GM) that commands the Monsters and sends Heroes on Quests..
multiple players that take on the roles of characters in the game (PC)
Timmy always screaming, "Death to the Orcs" whenever he rolls for an attack.
Board games can be role playing games and vice versa. Most board games have you take on the role of something/someone, be it a Gingerbread Man, a real estate investor, or island Settler. Traditionally, what makes a role playing game different than a regular board game are:
a campaign. Most role playing games have you play the same character over multiple gaming sessions. Most board games are played in a single sitting, where the outcome of previous games has no bearing on the current game.
variable rules. Many a GM will give a +2 circumstance bonus for clever play. GM's have been know to modify encounters from the rules to take it easier on a party of adventures that have been having bad luck with the dice, so they won't die as easily. Most traditional board games have a strict, rigid rules structure, and players are not encouraged to usually decide rules on the fly.
open objectives. Although a GM might have a particular campaign in mind, the players themselves shape the outcome of the story. Perhaps you want to go after the man who killed your father to flesh out your back story, and no amount of prodding from the GM can get you to save the village from the pending goblin horde. GM's and the players decide what the goals for their characters are.
story telling. Although I would guess many a GM, due to player interest, deemphasize the acting and "role" playing of the characters for a more hack'n'slash approach. Most board games don't have you speaking in accents, and trying to emulate conversations between Non-Player Characters.
There are of course exceptions to every difference. Some role playing games are becoming more like tactical board games. Some board games have campaigns (Descent/Risk Legacy), reward creative play outside the defined rules (Story Cubes), or have you acting out your role (Pitch). Word definitions change, so if people are calling something a board game, it probably is (at least to that person).
Definitions are only useful as long as they are useful. It is much more important that a game category reasonably describes aspects of the game that the person you are communicating with can understand, than that it can be summarized using a few technical bullets. Thus, when normal usage of a term conflicts with some suggested definition, then the definition is wrong, not the usage.
In other words, a role playing game is not a board game, because most people understand the term board game to exclude role playing games. Of course, there is always a gray zone, where a game could realistically be described as either category.
When playing a role-playing game, you spend about 80% of your time NOT using (or referencing to) the game rules.
Try not using the rules in a board game for even a little time...
