My mum has always equated intelligence/education/"breeding" with RP/plummy voices and, even though I know this is ridiculous, I haven't been able to shake off feeling uncomfortable about my slightly Cockney "twang".
She's not alone. Research has shown that some accents affect people's chances of success - in interviews, career progression, etc., because of the way their accents are perceived. I've noticed that TV dramas often reinforce these stereotypes - cockneys are very often the "villain" of the piece, the person not to be trusted, the less intelligent person, etc., etc. Similarly, certain rural accents are used for comedy effect.
As for grammar, like lots of people I feel it is important that people know the difference between "formal" and "informal" speech because, however unfairly, the way you speak does affect your chances in life.
I was listening to "Woman's Hour" the other morning when the subject of "gangsta" type language was being discussed and some young women from Hackney were interviewed. They were all very articulate and made some quite insightful comments about how language is constantly changing. They also saw the way in which new words and phrases come into being as being a creative process. It certainly gave me food for thought and made my think about my own knee-jerk reactions.