There are a lot of things to say about this topic, but I’ll try to keep it short. It can be helpful to understand the basic physiology of the fear response in order to understand what sets them apart from Panic Attacks, and why they feel the way they do.
The Autonomic Nervous System (ANS) takes care of your involuntary bodily functions (e.g., breathing, heart rate, pupil dilation etc.) by communicating signals from your brain to the rest of your body. Within the ANS, there are two major systems at play: the Parasympathetic Nervous System, which controls your body when you're calm, and the Sympathetic Nervous System, which takes over when you're excited or stimulated. When we experience a fear reaction, the sympathetic nervous system kicks into gear. It tells your body to produce adrenaline, your heart rate increases, you start to sweat, your muscles get tense. In a dangerous situation, all the adrenaline and hyper-vigilance would help you to defend yourself against a threat (fight), or to escape from it (flight). The manifestations of this physiology are intense, but normally you’re focused on a threat and not on your body’s responses, and normally they don’t last much longer than the threat. Normally, after the stimulus/threat/etc. is gone, the parasympathetic nervous system takes over and the body's heightened state returns to normal. During a panic attack, this second stage does not occur. Instead, you’re in a prolonged panic response with nothing to fight or flee from, your body isn’t calming itself down, and you become overwhelmed by your body’s response.
Your body and brain are reacting as though they’re in a life-or-death situation. Different people experience panic attacks differently, and different coping strategies can be very helpful to reduce the severity and duration of panic attacks.