There are two passages in the New Testament that mention an angel, first in Matthew 1.20, where the angel appears to Joseph sleeping, and also in www.ellopos.net/elpenor/greek-texts/new-testament/luke/1.asp" target="_blank">Luke 1.11, where the angel strengthens the Christ. The English translation renders both passages as "An angel appeared," while the Greek uses two different words:

ὤφθη δὲ αὐτῷ ἄγγελος ἀπ' οὐρανοῦ ἐνισχύων αὐτὸν (www.ellopos.net/elpenor/greek-texts/new-testament/luke/1.asp" target="_blank">Luke 1.11)

ταῦτα δὲ αὐτοῦ ἐνθυμηθέντος ἰδοὺ ἄγγελος Κυρίου κατ' ὄναρ ἐφάνη αὐτῷ (www.ellopos.net/elpenor/greek-texts/new-testament/matthew/1.asp" target="_blank">Matthew 1.20)

What exactly is the difference in meaning of the verb "appeared" in these two passages? Can we consider ἐφάνη as a concrete, sensory appearance in this case (Joseph is sleeping), and if so, what makes it different from the apparition of the angel in Luke?

It's great that you pay this attention to such differences. Let me add to your observations the fact that the first (ὤφθη) is used much more than the second (ἐφάνη) in the New Testament, which means, as I will explain immediately, that the authors of the New Testament are very interested in the personal experiences of faith, the condition of one's soul, the inner feelings and thoughts, more than in the outer, objective, facts.

To a great extent the verbs express the same meaning, which is the reason the translator renders both of them as "appeared". This is not wrong, it just lacks a subtlety that is necessary for someone who investigates deeper the meaning of Scriptures.

Beyond the common meaning of these verbs, ὤφθη regards mostly those who see, while ἐφάνη regards mostly those that reveal themselves. In one and the same reality, the reality of the appearance / revelation, ὤφθη regards mainly the perception of this appearance by those that were present when this appearance occured, and ἐφάνη regards mainly the side of the revealed, the fact that someone or something was revealed.

Applying these to your sentences, ὤφθη δὲ αὐτῷ ἄγγελος ἀπ' οὐρανοῦ ἐνισχύων αὐτὸν (www.ellopos.net/elpenor/greek-texts/new-testament/luke/1.asp" target="_blank">Luke 1.11), and ταῦτα δὲ αὐτοῦ ἐνθυμηθέντος ἰδοὺ ἄγγελος Κυρίου κατ' ὄναρ ἐφάνη αὐτῷ (www.ellopos.net/elpenor/greek-texts/new-testament/matthew/1.asp" target="_blank">Matthew 1.20), we can say that in the first case the emphasis is on the fact that he saw the angel, while in the second sentence the emphasis is on the fact that the angel came to him.

Therefore it should not be considered as a coincidence that in the case of Mat. 1.20 the text does not say much about the reaction of Joseph towards the angel, while Luke 1.11, describes in a lot of detail the condition of Zachary's soul when the Angel appeared.