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If you don't have the appropriate skill, you'd roll a d4. (p. 11 of the Core Set Rulebook.)
For most Spells it means you can Discard them instead of Banishing them.
As for the skill/die, that will depend upon the circumstances and which boons might be played to help. As Parody says, you would be using d4 since you don't have the skill, only the proficiency. If someone plans to play a boon that would improve your Arcane or Divine (but not both) check, you would go with that check (unless you didn't want to succeed at the check for some reason). Also, since you have both the Arcane and Divine proficiencies without either being associated with a skill like Intelligence or Wisdom, boons that apply to Intelligence or Wisdom wouldn't help unless they also apply to Arcane/Divine.
The primary benefit of having this power, though, is exactly as Parody says - the ability to discard, and potentially heal, a spell instead of banishing it. In addition, such spells might be available for rebuilding decks at the end of the scenario, in which case any player might keep the card (assuming you're not in society/organized play).