One significant difference between the Ordinary and the Extraordinary Forms is in the lectionary: the Ordinary Form uses a three-year cycle of Sunday readings and a two-year cycle of daily readings, which were newly arranged for the 1970s Missal; the Extraordinary Form contains a one-year cycle of readings, which had developed over the centuries with the Mass. The new lectionary was not composed, moreover, simply as a supplement to the old: it is a new composition, and the readings are different as or more often than they align.
One question that has persisted for many years is, “may one say the Extraordinary Form with the Ordinary Form readings?” At one point, the thought was, “yes, you may.” The Pontifical Commission Ecclesia Dei issued a letter in 1991 approving of the practice. Since then, however, the PCED has changed that ruling: the readings, calender, etc. of the new missal may not be substituted for those of the old.
For a comparative table of aggregate statistics for the two lectionaries, see this page.
To read the Mass readings themselves for any given day of the year, visit the Divinum Officium site, type a date in the appropriate box, and click “Sancta Missa” (this will give you the whole Mass, proper and ordinary, for the day).
That’s some basic information, but what about the background on the old lectionary, and information on why the new diverges from it so much? Probably the best one-stop resource for the old lectionary is the International Federation Una Voce’s 2013 position paper on the topic. The FIUV paper discusses, for instance, the great antiquity of the old lectionary and its development; it is excellently footnoted. It also discusses the fact that traditionally the liturgical context for expounding the Bible at length was in the Office, particularly at Matins. (The traditional office of Matins, or any other hour, for any day can be found at the Divinum Officium site as well.)
For a somewhat briefer companion to or summary of the FIUV paper, I suggest this blog post by Joseph Shaw, the president of the Latin Mass Society of England and Wales. There is also this post from a few weeks ago, excerpting a discussion of the old lectionary’s mystagogical ends versus the new lectionary’s pedagogical ends.
Finally, in the category of anecdotes, there is this first-hand reminiscence from Alfons Cardinal Stickler, who worked at the Second Vatican Council as a peritus:
In the Conciliar Constitution the introduction of a three-year Lectionary is nowhere spoken of. Through it the reform commission made itself guilty of a crime against nature. A simple calendar year would have been sufficient for all wishes of change. The Consilium could have stuck to a yearly cycle, enriching the readings with as many and as varied a choice of collection as one would want without breaking up the natural yearly course. Instead, the old order of readings was destroyed and a new one introduced, with a great burden and expense of books, in which as many texts as possible could be accommodated, not only from the world of the Church but also—as was widely practiced—from the profane world.[*] Apart from the pastoral difficulties for parishioners’ understanding of texts demanding special exegesis, it turned out also as an opportunity—which was seized—to manipulate the retained texts in order to introduce new truths in place of the old. Pastorally unpopular passages—often of fundamental theological and moral significance—were simply eliminated. A classic example is the text from 1 Cor. 11:27-29: here, in the narrative of the institution of the Eucharist, the serious concluding exhortation about the grave consequences of unworthy reception has been consistently left out, even on the Feast of Corpus Christi. The pastoral necessity of that text in the face of today’s mass reception without confession and without reverence is obvious.
That blunders could be made in the new readings, above all in the choice of their introductory and concluding words, is exemplified by Klaus Gamber’s note on the end of the reading on the first Sunday in Lent of the Reading Cycle for Year A, which speaks of the consequences of Original Sin: “Then the eyes of both of them were opened, and they knew that they were naked.” Whereupon the people, performing their duty of lively and active participation, must answer: “Thanks be to God.”
* His Eminence appears to be speaking here of the practice that arose in the foment during and after the Council, of substituting any old book for the Scriptures. The Consilium, of course, never recommended, and the Holy See never promulgated, any lectionary including Mass readings from books other than Scripture, although the ideas proposed by the Consilium’s members extended well into the range of the bizarre.