In general, (english has a lot of exceptions) 'better' is a quality and 'more' is a quantity. You have better hair than I do. I like your hair more than mine. archimed gives a good example. "I like him more" means you like the person. "I like his reading more" refers to the reading (your intent). He has better reading than the others. An ocean has more water than a lake. Lake water tastes better than ocean water. You know more languages than I. Your accent is better than mine.

"Better" and "more than" are often interchangeable. You have a few errors in your paragraph, so here's another way to write it. "There is a group of people who are reading out loud, and I think that one person reads better than the others. Can I say "I like him better" (yes, you can), implying that his reading skills are better. Are these sentences grammatically correct or just acceptable? Can I use them interchangeably? What's the difference? Notice that "group of people is singular. "... one person do it better..." Here you need "does" instead of "do." Again, it is singular. Some examples: "I like my boyfriend better than your boyfriend." "I like my boyfriend more than your boyfriend." Note: Above, you are comparing one person against "the others." This means you are doing a comparison of two things. You can also say, "I like him best," if you are comparing him with each of the other people (superlative). For you boyfriend, you can say, "I like my boyfriend best." This means you are comparing your boyfriend against everyone else's. If you have any questions, please let me know. :-)

I think you can improve this sentence like this: There is a group of persons which is reading out loud. I think that one of them reads better. I am impressed by his accent. I like it. As you aware, in English you should use short sentences.