Nice clarification, pjt33!
I'd just add that whether is used to introduce a series of expected alternatives (even if some of the alternatives are implied). When if is used, none of the situations under consideration have to occur.
Leveraging pjt33's example:
Ask him whether he likes peas has an implicit "or not" in it:
Ask him whether he likes peas (or not).
Ask him whether (or not) he likes peas.
I hope I can make this more clear via examples that give choice between two positives:
Ask him whether he likes peas or corn.
The speaker is typically expecting a choice from the given alternatives, peas or corn (or maybe both).
Ask him if he likes peas or corn.
Here the speaker is allowing for the choice of peas, or of corn, or perhaps neither.
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