Simply, vel is inclusive and aut is exclusive. As Lewis and Short put it:
In general aut puts in the place of a previous assertion another, objectively and absolutely antithetical to it, while vel indicates that the contrast rests upon subjective opinion or choice; i. e. aut is objective, vel subjective, or aut excludes one term, vel makes the two indifferent.
[Vel] As disjunctive conjunction, to introduce an alternative as a matter of choice or preference, or as not affecting the principal assertion (while aut introduces an absolute or essential opposition...
Going off your examples:
Licet tibi habere aut vim aut laetitiam.
This means, "You are allowed to have either power or happiness but not both."
Licet tibi habere vel vim vel laetitiam.
"You are allowed to have power, or perhaps happiness, or both, should you wish."
The enclitic -ve is the same as vel.
Bennett's example I think is more illustrative:
quī aethēr vel caelum nōminātur, which is called aether or heaven
That is, both aether and caelum as the name for the sky is correct.
So in your example, aut is the best word to use.