I've read about some of the differences, but they may be opinions not supported by best scientific evidence.
I think the limbic system, in particular amygdala and hippocampus would be activated in both, and in many other emotional reactions.
But even more relevant, given that this is cog-sci exchange, is that there is the idea that fear and anger are cognitively similar in that they both focus and narrow the mind, but that anger focuses the mind much more because fearing a certain object also makes one hypervigilant so the person could potentially react to lots of other potential signs of danger at the same time. But my thinking is that anger can also lead to arousal and person being irritated by lots of other previously-neutral stimuli.
Another idea I remember is that apparently fear leads to feeling cold in the extremities but anger the head and particularly the arms can remain or get warmer. Or as this NPR article says, fear leads to feelings in the chest area, but anger is one of the few feelings that activates the arms.
Internally it is my own experience that fear leads to shrinking and weakness while anger leads to a kind of activation and strength.
Externally there are behavioral differences but also in terms of body language and facial expression, apparently one can tell anger and fear apart. Clenching jaws and fists in anger...
But the problem is people rarely express these emotions so fully in their body or their face.
Or to be more accurate, it depends on the person and how expressive they are and how frightened/angry they feel, culture, age, etc. But such facial expressions as ones below are not ones I see around me often.
Lastly, I once had a talk with a doctoral candidate and very briefly we touched on this, which is obviously quite a big topic, but he told me one of the difficulties is that the feelings that follow each other very fast and also that a person might react to feelings with other feelings. For instance, you become angry at your child but then almost instantly you might feel fear at how angry you are, etc.
So the above are sort of what I've read or remember about the differences. I really like to learn more about this so if you feel this question is too broad, just choose the area that you know most about and correct my thinking there. Since this was the only psychology section of the forum, I posted it here, and surely anger/fear distinction not only can be studied by different branches of psychology (biopsych, behavioral psych, cog psych, clinical psych, etc), but as a human emotion, it can also be studied by anthropology, for instance, on the cultural level. So I'm aware of that.