以为 and 认为 means same in classical chinese but different in modern 以为 is something you thought would happen but didn't happen i.e 我以为他会来 in fact he didn't come 认为 is something you think will happen but doesn't happen yet i.e 我认为他会来 in fact he doesn't come yet but you think he will come when you want to express your opinions, you should say 认为

以为: the things happened are not like you think before 认为: just in your mind ! you think something will happen

It's quite easy. 以为 = thought 认为 = think, keep thinking

they all mean" i think". but 认为has stronger tone than 以为.i think Maple's answer is very good.觉得 is something that you are not sure,maby he will come,maby not

"认为" is much more sure than "以为". means "I belive, I feel" . 以为 contains meaning of "I guess" but actually not happen or not true. 我认为他是个伟大的科学家。He is a great scientist indeed. 我以为他是个伟大的科学家。 I think he is a great scientist at first, but now I don't think so!