Anger is an emotion. The conscious experience of the emotional reaction is the feeling.
What does that mean?
According to the American Psychological Association (APA), emotion is defined as "a complex reaction pattern, involving experiential, behavioural and physiological elements."
This define helps to explain why we can be angry but not feel angry. Have you heard of resting angry face? Someone looks angry, but when you ask them, they say they are not. There is a good chance they are experiencing the emotion of anger but are not consciously feeling of anger.
One of the reasons someone denies being angry when the people around them can see it clearly is because they are currently unable to feel their anger.
A separation of emotions and feelings is normal. Many people live in their heads and don't feel what is going on in their bodies. When growing up, the person was not taught to name their feelings. And in some cases, they were taught to deny feelings.
To help understand anger better check out the following chapters.
Growing up, did your parents stop you from crying? Stop you from feeling a certain way?
Were you told that you don't have anything to be angry/upset/sad about?
Were you told to be quiet and not make noise?
Were you told not to express your feelings?.