There are no mistakes in the Douay-Rheims Bible. It is a direct, word by word translation from the Latin. It has some archaic wording, including many of the book titles, and the psalm numbering is different, but those aren't errors.
Both are "translations of the Latin Vulgate", so in fact they claim to represent the same original text. The D-R and Knox have two different purposes, however.
The Douay is a literal translation. The Knox is a literary translation.
This means the D-R means to provide a word-for-word transliteration from Latin into 16th century English. Msgr. Knox, however, was trying to give the sense of the text in mid-20th century English ideas.
Understood that way, you can see, neither is "better" but both serve well for certain purposes, and poorly for others. The D-R is great if you want to understand the meaning of the Latin word that is translated, and good if you want to do apologetics, since its verbiage isn't paraphrasing the Latin, but literally translating. For reading Scripture in a setting for meditation, the Knox would be a far better choice, since it is designed to be a beautiful literary work, which is more apt to engender images for contemplation.
Given this, we can explain several of the odd phrases in the English of the D-R, especially in the Psalms, versus the Knox.Psalm 15 (16). 3-4Latin :
Sanctis qui sunt in terra ejus, mirificavit omnes voluntates meas in eis. Multiplicatæ sunt infirmitates eorum : postea acceleraverunt. Non congregabo conventicula eorum de sanguinibus,
nec memor ero nominum eorum per labia mea. D-R :
To the saints, who are in his land, he hath made wonderful all my desires in them. Their infirmities were multiplied: afterwards they made haste. I will not gather together their meetings for blood offerings: nor will I be mindful of their names by my lips. Knox :
There are faithful souls in this land of his; wondrous delight he gives me in their companionship. What do they do but lay up fresh store of sorrows, that betake themselves to alien gods? Not with these will I pour out the blood of sacrifice; I will not take forbidden names on my lips.
The Latin itself is pretty obscure here, because if you look at verses 1-4 there seem to be several independent thoughts going on. It's also hard to keep up with the personal pronouns being used. In two verses here we have four different personal pronouns. If they all refer to the same thing, the text makes no sense—The "saints" who are wonderful are suffering, getting faster, then meeting for blood offerings, and not to be even spoken of by the just psalmist? Hardly. The D-R does not try to make sense of it, just translates.
Knox has tried to coordinate the phrases so they follow a thread. In doing so he has to add expressions which could be implied but are missing. Whether that's the real original meaning, who can say for certain, but it is one logical reading. The psalm opposes faithful souls vs. those which are not faithful and unlike the psalmist and saints don't confess God as God. Thus it's the saints who are wonderful for the psalmist, but the unfaithful who increase their wickedness (by idolatry) and the psalmist will have no part with these pagans, nor even mention the idols' names.Psalm 76 (77). 18-21Latin :
Multitudo sonitus aquarum; vocem dederunt nubes. Etenim sagittæ tuæ transeunt; vox tonitrui tui in rota. Illuxerunt coruscationes tuæ orbi terræ; commota est, et contremuit terra. In mari via tua, et semitæ tuæ in aquis multis, et vestigia tua non cognoscentur. Deduxisti sicut oves populum tuum, in manu Moysi et Aaron.D-R :
Great was the noise of the waters: the clouds sent out a sound. For thy arrows pass: The voice of thy thunder in a wheel. Thy lightnings enlightened the world: the earth shook and trembled. Thy way is in the sea, and thy paths in many waters: and thy footsteps shall not be known. Thou hast conducted thy people like sheep, by the hand of Moses and Aaron.Knox :
How the waves roared, how the clouds volleyed rain, what echoes from their midst! To and fro thy arrows passed, thy crackling thunders rolled, till all the world shone with thy lightning, and the troubled earth shook. Thy way led through the sea, the deep tide made a road for thee, and none may read the traces of thy passage, where thou, with Moses and Aaron for thy shepherds, didst bring thy people out on their journey.
It should be obvious that the poetic style of the Latin imagery isn't well translated in the D-R, but the literary style is reproduced in Knox. One of the worst is this "thunder in a wheel". The image is of the rolling clap of thunder, so "thy crackling thunders rolled" is a far better translation. The odd plurals like the "paths" in "many waters" also don't accurately reflect in English the parting of the Red Sea, and a casual read in the D-R would have trouble identifying this psalm with the Exodus until the very last verse.