I too support the line of thinking that favors simply 'Something has come up,' along with a sincere apology.
I'm terribly sorry / so sorry, but ...
... something (important / urgent / unavoidable / unexpected) ...
... something I can't get out of ...
... has come up.
I'm afraid I ...
... can't make it after all.
... will have to cancel.
... need to reschedule.
(Or to explain why you missed an event, it could all be in the past tense: 'Something came up.')
I wouldn't use the word 'appointment,' because if you have already accepted one appointment, you really can't politely cancel it in favor of a different appointment. As Gibson mentioned, the rule of etiquette is that the first acceptance takes precedence, except in case of something like illness of yourself or a family member.
'Something got in the way' might not be completely wrong, but it seems less idiomatic to me. In any case, it would be so casual that you would only use it with a peer who was also a friendly acquaintance, someone you work with or meet with frequently, a family member, etc.
Anything with 'between' sounds too much like Denglish to me. In fact, 'got in the way' might also have 'dazwischengekommen' in the back of the mind, but that's just not how we express the idea.