How different would it be

Probably not at all. The issue of the Carter's ill-gotten riches was really just a secondary issue in the novel. It explained reverend Sullivan's antipathy towards Landon (initially) and his family, but was in no way nearly as important as the story's central theme: the budding relationship between Landon and Jamie resulting in their falling in love, followed by her untimely demise. Including the back story of their wealth in the movie would just have been a distraction and there's only so much that can be added to a 100 minute film.

That's the main problem in trying to make a movie out of a novel - there's just too much detail that has to be left out and/or glossed over due to time constraints. It's one of the reasons I almost always dislike the movie version when I've previously read the book. However, in this case, I saw the film first and then read the story. Though they are told completely differently, it didn't bother me a bit - I think both stand well on their own merits.