An arc is first of all a curve that is made up of part of the circumference of a circle. So it has a particular form which may well be like that of an arch, but it’s always part of that circumference in this case. It can also be the electrical discharge that jumps from one point to another (so lightning forms an arc: not in this case a nice tidy bit out of a circumference) and finally we have the metaphorical use in a “story arc” in a fiction book, film, TV series or play (often across several episodes of a TV series) which traces the development of a plot or side plot. The verb to arc means to move with a curving trajectory, which could include arching over something.
An arch is a physical thing rather than a mathematical concept or a plot device (though you can have over-arching ideas that act as a sort of umbrella across a narrative or other story). So it’s a symmetrical curved (though that curve can be quite pointy) structure that supports a bridge, a wall, etc. It’s also the inner side of the foot, which is the same thing but in nature rather than constructed. The verb means to form an arch.
So an arc has a specific shape unless it doesn’t, and an arch is a physical thing unless it’s a metaphor. But you don’t have a story arch and most arches couldn’t be said to form part of a circumference of a circle.