What are the main differences between the Catholic and Protestant definitions of Biblical inerrancy?

There is not any real difference between the Catholic teaching on inerrancy of Scripture and the generic "Protestant" view given in this answer [which has since been deleted]. The two definitive documents in recent history are probably Pope Leo XIII's encyclical Providentissiumus Deus and the Second Vatican Council Document Dei Verbum. These points more or less summarize the position given in the question description.

• All Scripture is inspired by God, having God for its author, and therefore can contain no error.
• Scripture truly has both a single Divine author and many individual human authors.
• There may be errors in copies due to errors in transmission, but such errors do not extend to the original inspired copies.

This is the most relevant material from PD:

20. The principles here laid down will apply to cognate sciences, and especially to History. It is a lamentable fact that there are many who with great labor carry out and publish investigations on the monuments of antiquity, the manners and institutions of nations and other illustrative subjects, and whose chief purpose in all this is too often to find mistakes in the sacred writings and so to shake and weaken their authority. Some of these writers display not only extreme hostility, but the greatest unfairness; in their eyes a profane book or ancient document is accepted without hesitation, whilst the Scripture, if they only find in it a suspicion of error, is set down with the slightest possible discussion as quite untrustworthy. It is true, no doubt, that copyists have made mistakes in the text of the Bible; this question, when it arises, should be carefully considered on its merits, and the fact not too easily admitted, but only in those passages where the proof is clear. It may also happen that the sense of a passage remains ambiguous, and in this case good hermeneutical methods will greatly assist in clearing up the obscurity. But it is absolutely wrong and forbidden, either to narrow inspiration to certain parts only of Holy Scripture, or to admit that the sacred writer has erred. For the system of those who, in order to rid themselves of these difficulties, do not hesitate to concede that divine inspiration regards the things of faith and morals, and nothing beyond, because (as they wrongly think) in a question of the truth or falsehood of a passage, we should consider not so much what God has said as the reason and purpose which He had in mind in saying it -- this system cannot be tolerated. For all the books which the Church receives as sacred and canonical, are written wholly and entirely, with all their parts, at the dictation of the Holy Ghost; and so far is it from being possible that any error can co-exist with inspiration, that inspiration not only is essentially incompatible with error, but excludes and rejects it as absolutely and necessarily as it is impossible that God Himself, the supreme Truth, can utter that which is not true. This is the ancient and unchanging faith of the Church, solemnly defined in the Councils of Florence and of Trent, and finally confirmed and more expressly formulated by the Council of the Vatican. These are the words of the last: "The Books of the Old and New Testament, whole and entire, with all their parts, as enumerated in the decree of the same Council (Trent) and in the ancient Latin Vulgate, are to be received as sacred and canonical. And the Church holds them as sacred and canonical, not because, having been composed by human industry, they were afterwards approved by her authority; nor only because they contain revelation without error; but because, having been written under the inspiration of the Holy Ghost, they have God for their author." Hence, because the Holy Ghost employed men as His instruments, we cannot therefore say that it was these inspired instruments who, perchance, have fallen into error, and not the primary author. For, by supernatural power, He so moved and impelled them to write -- He was so present to them -- that the things which He ordered, and those only, they, first, rightly understood, then willed faithfully to write down, and finally expressed in apt words and with infallible truth. Otherwise, it could not be said that He was the Author of the entire Scripture. Such has always been the persuasion of the Fathers. "Therefore," says St. Augustine, "since they wrote the things which He showed and uttered to them, it cannot be pretended that He is not the writer; for His members executed what their Head dictated." And St. Gregory the Great thus pronounces: "Most superfluous it is to inquire who wrote these things -- we loyally believe the Holy Ghost to be the Author of the book. He wrote it Who dictated it for writing; He wrote it Who inspired its execution."

21. It follows that those who maintain that an error is possible in any genuine passage of the sacred writings, either pervert the Catholic notion of inspiration, or make God the author of such error. And so emphatically were all the Fathers and Doctors agreed that the divine writings, as left by the hagiographers, are free from all error, that they labored earnestly, with no less skill than reverence, to reconcile with each other those numerous passages which seem at variance -- the very passages which in great measure have been taken up by the "higher criticism;" for they were unanimous in laying it down, that those writings, in their entirety and in all their parts were equally from the afflatus of Almighty God, and that God, speaking by the sacred writers, could not set down anything but what was true. The words of St. Augustine to St. Jerome may sum up what they taught: "On my part I confess to your charity that it is only to those Books of Scripture which are now called canonical that I have learned to pay such honor and reverence as to believe most firmly that none of their writers has fallen into any error. And if in these Books I meet anything which seems contrary to truth, I shall not hesitate to conclude either that the text is faulty, or that the translator has not expressed the meaning of the passage, or that I myself do not understand."

And from Dei Verbum:

11. Those divinely revealed realities which are contained and presented in Sacred Scripture have been committed to writing under the inspiration of the Holy Spirit. For holy mother Church, relying on the belief of the Apostles (see John 20:31; 2 Tim. 3:16; 2 Peter 1:19-20, 3:15-16), holds that the books of both the Old and New Testaments in their entirety, with all their parts, are sacred and canonical because written under the inspiration of the Holy Spirit, they have God as their author and have been handed on as such to the Church herself. In composing the sacred books, God chose men and while employed by Him they made use of their powers and abilities, so that with Him acting in them and through them, they, as true authors, consigned to writing everything and only those things which He wanted.

Therefore, since everything asserted by the inspired authors or sacred writers must be held to be asserted by the Holy Spirit, it follows that the books of Scripture must be acknowledged as teaching solidly, faithfully and without error that truth which God wanted put into sacred writings for the sake of salvation. Therefore "all Scripture is divinely inspired and has its use for teaching the truth and refuting error, for reformation of manners and discipline in right living, so that the man who belongs to God may be efficient and equipped for good work of every kind" (2 Tim. 3:16-17, Greek text).

12. However, since God speaks in Sacred Scripture through men in human fashion, the interpreter of Sacred Scripture, in order to see clearly what God wanted to communicate to us, should carefully investigate what meaning the sacred writers really intended, and what God wanted to manifest by means of their words.

To search out the intention of the sacred writers, attention should be given, among other things, to "literary forms." For truth is set forth and expressed differently in texts which are variously historical, prophetic, poetic, or of other forms of discourse. The interpreter must investigate what meaning the sacred writer intended to express and actually expressed in particular circumstances by using contemporary literary forms in accordance with the situation of his own time and culture. For the correct understanding of what the sacred author wanted to assert, due attention must be paid to the customary and characteristic styles of feeling, speaking and narrating which prevailed at the time of the sacred writer, and to the patterns men normally employed at that period in their everyday dealings with one another.

But, since Holy Scripture must be read and interpreted in the sacred spirit in which it was written, no less serious attention must be given to the content and unity of the whole of Scripture if the meaning of the sacred texts is to be correctly worked out. The living tradition of the whole Church must be taken into account along with the harmony which exists between elements of the faith. It is the task of exegetes to work according to these rules toward a better understanding and explanation of the meaning of Sacred Scripture, so that through preparatory study the judgment of the Church may mature. For all of what has been said about the way of interpreting Scripture is subject finally to the judgment of the Church, which carries out the divine commission and ministry of guarding and interpreting the word of God.

13. In Sacred Scripture, therefore, while the truth and holiness of God always remains intact, the marvelous "condescension" of eternal wisdom is clearly shown, "that we may learn the gentle kindness of God, which words cannot express, and how far He has gone in adapting His language with thoughtful concern for our weak human nature." For the words of God, expressed in human language, have been made like human discourse, just as the word of the eternal Father, when He took to Himself the flesh of human weakness, was in every way made like men.