Although fear has been classified as an emotion by psychologists, it is a very basic human emotion and can be almost considered as a simple feeling. In fact if emotions comprise of feelings and bodily reactions, then fear would be the basic feeling component of anxiety or phobias as has been explained in the psychology of emotions. I prefer the use of fear as feeling rather than as an emotion and to explain this, it is important to distinguish between feelings and emotions in psychology. As of now this distinction is blurred and psychologists do not distinguish between feeling and emotion extensively.
Emotions are complex mental and physical processes as emotions involve feelings which are mental or psychological components and bodily reactions which are physical reactions. Thus feeling is an essential part of emotion. The simple feeling is purely psychological and does not involve bodily reactions and that is how fear which may or may not involve bodily reactions could be both a feeling and an emotion. For example fear of a student at the examination centre would be accompanied by bodily reactions such as rapid heart beats or flushed face, dilated pupils and so on. Although fear which could be a feeling component may be very simple and generalized and could be even unconscious without the presence of bodily reaction, although not as intense as anxiety which necessarily involves bodily reactions. Thus perhaps you are on stage and performing a play, you may not directly sense any bodily reaction and could be calm and normal but you may still harbour a feeling of fear as in some sort of uneasiness.
Fear could thus be both a feeling and an emotion yet fear as an entirely subjective or mental feeling component would be difficult to detect as it would not be accompanied by visible or noticeable physical reactions as in fear as an emotion. Anxiety on the other hand is considered a distinct internalized emotion as it arises internally from a perceived threat rather than fear which is due to external stimuli. Fear could be defined as an externalized emotion or feeling which may or may not be accompanied by bodily reactions and fear could be conscious or unconscious.
A psychology of fear would distinguish fear as an emotion and fear as a feeling, fear as conscious and fear as unconscious as well as fear with bodily reactions and fear without bodily reactions and fear in anxiety and fear in phobias. It would be important to understand why fear occurs and what are the bodily reactions when fear is a strong conscious emotion and how this differs from fear as a feeling which may not have bodily reactions and could be conscious but would more likely would be unconscious.
Say for example you have an unconscious fear of old haunted dilapidated houses and you repeatedly dream of events in such houses, the dream itself could cause some bodily reactions but it is not apparently obvious that the fear is causing the bodily reaction. So in this case, the fear itself is simply unconscious and a feeling which manifests in dreams and the dreams are associated with bodily reactions and not the fear. So here fear is a feeling rather than an emotion. Some psychologists would however argue that this "fear' could simply be an undefined anxiety but since the psychologists would also identify the cause of the fear (or anxiety as they would say) which is a fear of haunted houses, this is still an externalized fear and not internalized anxiety. Fear is thus externalized whether it is a feeling or an emotion.
The distinction between feelings and emotions are like trying to distinguish between meteoroids and asteroids in outer space and the distinction would require very detailed analysis of the layers of the mind. At this point we do not have sufficient frameworks or scientific evidence that would help to make this distinction easy. Modern studies in consciousness have focused on this problem of feeling. As Thomas Nagel highlighted in his very famous paper "What is it like to be Bat?' this subjective feeling of "what it is like to be' is very important. With the study of consciousness, the subjective aspects of emotions are all important and although hardcore physicalists who believe that our minds are nothing but neuronal firings would ignore that an emotion would have a feeling aspect, studies in consciousness have proved that feeling or the subjective aspect of being is the core of being human.
However the topic being the psychology of fear, it is essential to understand why fear happens and what would be the therapeutic directions for fear. Fear could be characterized as a general unconscious feeling of unpleasantness or could be more complex emotion such as anxiety and externalized fear. Fear could also be phobias which are persistent pathological fears directed towards specific objects and situations. Phobias could be of spiders or heights and these are exaggerated or extreme forms of fear with severe bodily reactions, almost bordering on paranoid reactions. Fear could thus be revealed as severe emotions with bodily reactions, and can be manifested as anxiety or phobia, fear could also be a general feeling. In order to understand why fear happens we distinguish the types of fear:
Fear as Emotions - Fear when accompanied by bodily reactions would signify strong emotional response to a situation or an object or event. Since fear is internalized, fear would naturally begin with a feeling or a subjective component. This means the individual would first "feel' afraid of the situation and then react to it. Such fear in which the individual is conscious of the emotion and reacts strongly to it is generally an emotional response and this sort of fear is thus manifested as a strong emotion.