St. Jerome translated the Septuagint (a Greek translation of the Hebrew Bible) into Latin. Later, he translated the Hebrew Bible (i.e., the protocanonical books) into Latin. He didn’t bother with the deuterocanonical books because they were already translated as part of the translation of the Septuagint version.
For the most part, the Vulgate contains St. Jerome’s translation from the Hebrew. However, St. Jerome’s translation of Psalms from the Septuagint was very popular, especially among the Gauls. It was used in the liturgy and people were familiar with the way it sounded. Spain and a few other countries, however, preferred St. Jerome’s translation from the Hebrew. Ultimately, St. Jerome’s translation of Psalms from the Greek Septuagint rather than from the Hebrew Bible found its way into the Clementine Vulgate. The Douay-Rheims was a translation of the Clementine Vulgate, so it translates St. Jerome’s Latin translation of the Greek Septuagint when it comes to the book of Psalms rather than St. Jerome’s Latin translation of the Hebrew Bible when it comes to the book of Psalms.
In the Latin translation according to the Greek, the 22nd Psalm reads “Dominus regit me,” or “the Lord ruleth me.” In the Latin translation according to the Hebrews, the 23rd Psalm reads “Dominus pascuit me,” or “the Lord shepherds me.”
When the Nova Vulgata was issued, the revisers chose to correct St. Jerome’s Latin translation according to the Greek to bring it closer to the Hebrew text. I suspect they wanted to maintain the traditional phraseology as much as possible for liturgical reasons.
St. Jerome’s translation of Psalms from the Hebrew is actually much better than his translation from the Greek, and it’s a great pity that the Clementine Vulgate didn’t include the former rather than the latter.