While the "profanity" may not have added anything to the storyline per se it is, apparently, merely one possible technique used for "curing" a stammer and that was the primary struggle being depicted.
The effectiveness is questionable however...
"The King's Speech" also touches upon the more ineffective therapies used during the time period, which included trying to speak with pebbles in the mouth and cursing up a storm to get the words out.
Another lesson is anger can produce fluency for someone who stutters. Albert is nearly always fluent when he is angry and cursing. So am I.
The Stuttering Foundation
Lionel Logue: You don’t stammer when you swear.
Nevertheless, an alternate version of the film was issued.
...the Weinstein Co was told that, if 3 of the 5 uses of the swear word “Fuck” were muted, then the pic would receive a lesser rating than its current “R”. No film footage was altered in keeping with director Tom Hooper’s insistence that the stammering king’s pivotal therapy-by-cursing scene not be cut. Deadline initially broke the news that The Weinstein Co was seeking the lower PG-13 film rating in response to educational and church groups who wanted to show the movie.
This artistic, or rather commercial, license annoyed Firth, who on Oscar night told reporters, “I don’t support it. I think the film has integrity as it stands. I think that scene belongs where it is. I think it serves a purpose.” The cut version also stoked spasms of outrage in the college of critics. “For the benefit of a teensy portion of the population,” wrote John Serba of the Grand Rapids (Mich.) Press, “the artistic vision of director Tom Hooper is compromised …” Roger Ebert tweeted, “Today is the last day you can see ‘The King’s Speech’ with the F word. F**k!”
Also, the "R" rating was not used in the UK... where it was initially given a "15" rating which was then downgraded
..BBFC later announced that its classification decision had been given "careful consideration" by its president and director. The board took the view it would be suitable for the less restrictive 12A rating. The BBFC added: "The public would understand why the board has reached this decision."
Colin Firth, who plays the reluctant king, said the context of the words was everything – that they were not being used in a sexual way or to insult or describe anyone. "It would be very interesting to know who the people are who would complain about that stuff ahead of violence."