There may be someone on here who can give a more grammatically correct answer, but as a native speaker, here's my two cents: Think of a circumstances as an event that has a beginning, middle and an end. When you say a circumstance arises, there is more focus on the beginning of the event (how it started). When you say a circumstance occurs, then there is less focus on a specific part of the event, but it's talking about the event happening as a whole. I hope that makes sense. Here's some examples off the top of my head. If an emergency situation arises, you need to act in a decisive manner to make sure that the situation is resolved. When an emergency situation occurs, there is a chain of events that happens all the way up until the emergency is resolved. In most situations the words are definitely interchangeable, but this is just a tiny nuance that I could think of.

I would probably use "arises" for something that is unexpected or a good thing or a bad thing, something that is going to require a response/decision/change in plans. I might use occur for something more neutral. Example: If the opportunity arises, I may move to New York. If the situation arises, we may need to call in more help. If that occurs, just close the app and reopen it.