Not necessarily the same interests for all their lives, although there's often more consistency, but especially the type of interest and whether the interest is useful in some way other than satisfying a curiosity, whether it's goal-oriented, are crucial in distinguishing between a BP and a ASD obsession.

Music generally tends to be something that people with ASD are less interested in than those with BP (or, rather, for those with BP there is or there's more likely to be some real need; it fulfills a role and a possible obsession may therefore be more goal-oriented).

Learning to play an instrument is not necessarily a sign of either disorder. A desire to play music accurately and to approach music rationally might be.

Abstract thinking can be a sign of both, but the accuracy by which concepts/words are used might help to distinguish the thinking of someone with ASD from that of someone with BP, for example. Also consistency of thought (also more generally) might be used to make a distinction.

Depression (and loss of interest) isn't a sign of one disorder in particular, but the reasons behind the depression might be.

Loss of interest due to a sudden new interest, especially when based on (rather extreme) associative thinking, is more likely a sign someone could be described as having BP.