I can't recall reading your other answers, so please forgive if this has been covered elsewhere, but there are a couple of possibilities, the first that spring to my mind are how each is 'cast'.
1) Direct Divine intervention requires some form of request/prayer (i.e. "[insert name of god], in thy name [please do xxxx]")
2) There is a direct visual effect (i.e. a golden halo/glow appearing around the caster, the gods/spirits symbol appearing briefly)
3) Divine intervention/spells are only available to the truly faithful or devout, and may need some form of holy symbol to use as a focus (i.e. crucifix etc)
4) Magic requires the use of spell components and incantations. Material items that are used as to cast the spell but are consumed in the process (I think this used to be a requirement in D&D - too many years since I played it), whereas Divine magic doesn't.
5) Possibly, the divine intervention can only occur if the 'caster' is possessed by the God/Spirit that could leave a visually noticeable imprint on the caster, literally becoming a vessel of the divine will - eyes glowing, tattoos appearing that kind of thing.