Yes. Catholics do have different Bibles than non-Catholics. There is no denying that.
FROM THE TRANSLATORS TO THE READER IN THE ORIGINAL KJB
If we should tell them that Valla, Stapulensis, Erasmus, and Vives found fault with their vulgar Translation, and consequently wished the same to be mended, or a new one to be made, they would answer peradventure, that we produced their enemies for witnesses against them; albeit, they were in no other sort enemies, than as S. Paul was to the Galatians, for telling them the truth [Gal 4:16]: and it were to be wished, that they had dared to tell it them plainlier and oftener.
But what will they say to this, that Pope Leo the Tenth allowed Erasmus' Translation of the New Testament, so much different from the vulgar, by his Apostolic Letter and Bull; that the same Leo exhorted Pagnine to translate the whole Bible, and bare whatsoever charges was necessary for the work?... If they say, it was one Pope's private opinion, and that he consulted only himself; then we are able to go further with them, and to aver, that more of their chief men of all sorts, even their own Trent champions Paiva and Vega, and their own Inquisitors, Hieronymus ab Oleastro, and their own Bishop Isidorus Clarius, and their own Cardinal Thomas a Vio Caietan, do either make new Translations themselves, or follow new ones of other men's making, or note the vulgar Interpreter for halting; none of them fear to dissent from him, nor yet to except against him. And call they this an uniform tenor of text and judgment about the text, so many of their Worthies disclaiming the now received conceit? Nay, we will yet come nearer the quick: doth not their Paris edition differ from the Lovaine, and Hentenius his from them both, and yet all of them allowed by authority?
Nay, doth not Sixtus Quintus confess, that certain Catholics (he meaneth certain of his own side) were in such an humor of translating the Scriptures into Latin, that Satan taking occasion by them, though they thought of no such matter, did strive what he could, out of so uncertain and manifold a variety of Translations, so to mingle all things, that nothing might seem to be left certain and firm in them, etc.?
Nay, further, did not the same Sixtus ordain by an inviolable decree, and that with the counsel and consent of his Cardinals, that the Latin edition of the old and new Testament, which the Council of Trent would have to be authentic, is the same without controversy which he then set forth, being diligently corrected and printed in the Printing-house of Vatican? Thus Sixtus in his Preface before his Bible. And yet Clement the Eighth his immediate successor, publisheth another edition of the Bible, containing in it infinite differences from that of Sixtus, (and many of them weighty and material) and yet this must be authentic by all means.