A very noticeable effect is that fearful civilizations will usually agree to unfavorable trade agreements, and even agree to demands via the demand screen ("just give me gold"). Also, you can demand they will not settle cities next to you and they will comply.
Remember, civilizations will not indicate approval to unfavorable trade agreements when you click "what will make this work", but they will approve of the agreement when you click "propose", if they are fearful. They will say something like "well, seems like I have no choice".
Keep in mind that I have had fearful civilizations go directly to "hostile" once they got their military force strengthened.
It's very hard to trade with hostile civilizations, you really need to offer agreements that are unfavorable to you for them to agree. In particular, it's very difficult to get open borders from them, even if you offer is otherwise amazingly in their favor. Also, a hostile civilization is - in my experience - very likely to start a war with you if they think they have the upper hand. This can also happen with non-hostile civilizations, but is more pronounced for hostile ones.