The Angelus is composed of three versicle-response prayers, alternating with Hail Marys. The phrase in question is from the beginning. In Latin:
V: Angelus Domini nuntiavit Mariae.
R: Et concepit de Spiritu Sancto.
I learned this in English as
V: The angel of the Lord declared unto Mary.
R: And she conceived of the Holy Ghost.
If I were translating it into English de novo, I might come up with something like
V: An angel of the Lord made an announcement to Mary.
R: And she conceived by [the power of] the Holy Spirit.
The Latin de is a very common preposition that can mean "from", "of", "away from", "concerning", and sometimes other things. Here it is translated "of" and "by" in the two translations.
Note: The preposition "de" is used in the Latin translations of the Apostle's and the Niceno-Constantinopolitan Creeds to describe the conception of Jesus; in both cases, the preposition is used to translate the Greek ἐκ. A standard Greek dictionary, Liddell-Scott-Jones, has a vast number of possible translations or usages for the word—the most relevant ones are probably those having to do with describing place or person of origin (III. (4) and (6) in the LSJ definition list).
The English translation I learned probably dates back to the 19th century or before. Its use of "of" to mean "by" was customary in early modern English through at least the 18th century:
Was it well done of rash Virginius
To slay his daughter ...
(Shakespeare, Titus Andronicus, Act V, Scene 3)
For it is a shame even to speak of those things which are done of them in secret.
(Ephesians 5:12, King James Version)
Note here that for example the New American Bible Revised Edition translates this verse
for it is shameful even to mention the things done by them in secret
The phrasing appears later, in Biblical commentary:
...not that either the law of God, or Gospel of Christ, require this to be done of all men ...
(Gill's Exposition of the Whole Bible [written mid-1740s], commentary on Matthew 19:21)
Note that Gill in the same chapter uses "done by":
...Mark adds, "but not with God; for with God all things are possible"; to be done by him...
At the very least, however, this was not an uncommon usage; and I see it persisting even into the 19th century.
My conclusion, then, is that the two phrases mean the same thing—there is no theological difference between them—but one is characteristic of an older variety of English.