What is the difference between the language used by the Easterner and the Westerner in this story?
One of the themes of this story is the "culture clash" between the eastern and western parts of the United States during this time period. In those times (and to some degree still) the East was seen as an area of elites -- people with high culture and lots of education while the West was seen as an area of backwards people. These perceptions are shown in the language Twain uses in this story.
Right from the beginning of the story, you can see that the Easterner uses very formal, stilted language. He talks as if he were giving a speech or writing a formal essay. For example, let's look at the first line of the story:
In compliance with the request of a friend of mine, who wrote me from the East, I called on good-natured, garrulous old Simon Wheeler, and inquired after my friend's friend, Leonidas W. Smiley, as requested to do, and I hereunto append the result.
The words and phrases "in compliance with," "garrulous," "inquired after," "as requested to do," and "hereunto append" are very formal. They sound like someone who is highly educated and wishes to show it.
By contrast, the Westerners talk like backcountry hicks. Let's look at the following sentence:
There was a feller here once by the name of Jim Smiley, in the winter of '49 or may be it was the spring of '50 I don't recollect exactly, somehow, though what makes me think it was one or the other is because I remember the big flume wasn't finished when he first came to the camp; but any way, he was the curiosest man about always betting on any thing that turned up you ever see, if he could get any body to bet on the other side; and if he couldn't, he'd change sides.
Here, you have words like "feller" and "recollect" and "body" (meaning a person). These are uneducated usages. You also see that the sentence runs on and it digresses (it goes off and talks about the flume for a while). It does not sound at all educated the way the Easterner's long sentence does.
Twain is trying to create stereotypical Eastern and Western figures and so he uses exaggerated language to make the characters sound more stereotypical.