I'll watch for an authoritative answer as well. But my general understanding is that appeal is for traversing legal conclusions reached by the examiner and petition is for traversing procedure followed by the examiner (or the Office).
I cannot think of a better generalization. But understand that the generalization is just a thumbrule.
A few legal conclusions are not appealable to the board. Examiner decisions related to restriction/election are not appealable to the Board. Also look at the distinction made in 608.04(c) regarding decisions related to new matter.
Consider the situation where a 102(a) rejection is maintained by the examiner despite your attempts attempts to perfect foreign priority. Should you petition to supervise the examiner or should you appeal the 102(a) rejection to the Board? Can you do both? What happens if you err in choosing between petition/appeal?
Rejections are generally appealable while objections are generally petitionable. It follows from that rough rule that things governed by statute are generally appealable, but things governed only by the rules and the MPEP are generally petitionable. Of course some statutory issues do result in objections (e.g. improper multiple dependent claims).