Huck's dialect is reflective of Twain's use of regionalism and local color in his writing. He populates his work with characters who speak like the locals would have actually spoken in that area. This is much the same as modern-day writers who use urban slang in their works to give their characters a "street" quality or the use of "country" dialect to represent characters who live in rural areas. When a writer chooses to use improper grammar and phonetic spellings of words such as Twain does, the writer is making the characters more realistic. An uneducated youth from a small southern town is not going to speak like a Harvard educated English professor! Had Twain used "standard" English, he would not have achieved the same effect with his characters. Readers would have gotten to know them for what they said, but not for who they really were (or who they actually represented).
Yet Huck is still on a slightly higher level than his pap. Huck states:
If I never learnt nothing else out of pap, I learnt that the best way to get along with his kind of people is to let them have their own way.
He uses some regional dialect (learnt nothing else out of) but nowhere near as much as his "pap" uses:
Who told you you might meddle with such hifalut'n foolishness, hey?
says his pap about Huck's attempts at education. Could you imagine if his father had said: "Who encouraged you to think that the pursuit of education was within your reach?" for example, it just wouldn't sound the same!
His dialect is one of the South pre civil-war, and its effect is somewhat dependent on your perspective. Some would argue that it lends a certain authenticity to the novel, showing that the author is actually familiar with the customs and the speech of the time. It was also somewhat shocking to certain areas of society given that he used bad language and swear words, which was not entirely accepted at the time of the novel's publication.
One other possible effect of the dialect is to bring out the conflict and contrast between the perception of Huck as uncivilized and uneducated and his obvious intelligence as he navigates all kinds of difficult situations and problems. It lends a certain depth to the idea of how we judge people based on speech or appearance rather than their inner character or their actions.
If the novel were in standard English, it is difficult to imagine it would have some of the same effect. I think it would lose some level of authenticity and a certain affect that make it more interesting and meaningful. By removing the colloquialisms of the language, and using instead the more stratified and "accepted" English, I believe a great deal would be lost from the novel and its power as a descriptor of America at the time of its conception and publication.
There are all sorts of ways that Huck's dialect differs from regular English. He uses "incorrect" pronunciations of words and incorrect grammar or all sorts.
If the novel were written in Standard English, it would not hardly ring true. It would be very difficult to believe that a boy in that time and place who is almost completely uneducated would speak regular English. It would also be very difficult that an uneducated slave would speak proper English as well.
So the book would not seem as authentic as it does, in my opinion. There would also not be such a clear distinction between those who speak properly and those who do not.