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."
The board itself versus the Narrative
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.
Continuing Campaigns and Character Growth
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.
Speaking in Character.
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.
There's a clear line
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.