To belie is to create an impression that is inconsistent with the facts or that contradicts evidence. For instance, if a friend is feeling very nervous and insecure about a presentation but somehow manages to give off an aura of nonchalant confidence while delivering it, you could say their casual, relaxed manner belies their insecurity. The core idea of belie is one of misrepresentation, though the term does not usually suggest an intent to deceive; rather it used to point out notable or striking variances or contradictions.
If a person is open about their feelings on a topic, they speak freely and hold nothing back. When someone is described as candid, it usually means they are frank, outspoken, and sincere—sometimes unpleasantly so. A candid remark is one free from reservation or disguise. Candid is also used to refer to things that are unposed and informal, such as a candid photo.
To like something is to regard it with favor or to find it agreeable. To appreciate something is to value it highly or to be grateful for it. This verb suggests wise judgment, delicate perception, and keen insight in realizing the worth of something. You might appreciate (or be grateful) for a friend’s thoughtfulness or appreciate (regard highly) the nuanced flavors of a fine wine. Appreciate is also used to talk about grasping something mentally or being aware of it, as in the case of someone who appreciates (but is not grateful for!) the dangers of a situation.