"Appear" and "seem"

Solution 1:

Appears alludes to appearance, to visibility or visible, tangible evidence whilst seems has a more intangible, impressional nature. Nevertheless, it seems they tend to be used interchangeably or at least, that's how it appears to me.

Solution 2:

I think there is no significant, discernible difference in meaning.

However, there may be a difference in origin. English has many words of similar meaning that are either French/Latin origin or Germanic/Anglo-Saxon origin. Appear seems to be the Latin form.

As an aside, words of Latin origin were often used by the upper classes. While those of Germanic origin were often often associated with the masses.

Appear: Latin apparēre, from ad- + parēre to show oneself

Seem: Scandinavian origin; akin to Old Norse sœma to honor