All emotions are feelings, but not all feelings are emotions. Fear, sadness, happiness and envy, for example, are both emotions and feelings. But if something gives you a relaxing feeling, for example, this is a feeling but not an emotion.

Sorry, Tetiana, but that isn't right. You seem to have misunderstood the meaning of the word 'emotion', and you seem to have confused the outward expressions of emotions with the emotions themselves. Smiling and frowning are facial expressions: they are not emotions.

Feeling are what’s inside and not always shown; emotions are visual representation of feelings. For example smiling or laughing (which are emotions) when you’re happy (feeling), or frowning or crying (also emotions) when you’re sad (feeling). Basically you show your feelings by emotions.