If you go to the Vatican’s website and read Summorum Pontificum, you’ll find that the 1960 Roman Breviary has been definitely made available along with the 1962 Mass (remember, the Divine Office is a liturgical act, an extension of the Mass so to speak), so there should be no qualms about saying it; you are praying along with all your brothers and sisters who adhere to the usus antiquor throughout the world, you’re not isolated.
Now, as for the difference between the Roman Breviary and the Liturgy of the Hours, it is much the same kind of difference that exists between the 1962 and 1970 missals: i.e. the prayers are in fact very different in place, similar in others, but it is not simply a matter of different languages; the LOTH was first published in Latin before translation occured.
You’ll find that the older office conforms to the millenia old practice of reciting the psalms weekly, rather than the innovation of the four-week cycle that LOTH uses, so it takes a little more time to say it. Also, the old office follows the old liturgical calendar, so it will be in line with the readings for the day at the EF Mass, you will be in the season of Septuagesima now, not ordinary time, for example.
I hope this helps. An expert could do better but this should make the main outlines clear.
I wish you the best. Praying with the office has been one of the best things I ever decided to do.
Actually the psalter of Pius the X (1910) which is the one used for the 1960 Roman Breviary broke with many traditions. In particular the 3 “Laudate” psalms (148-148-150) were broken up at Lauds; for millennia they had been kept together as the last three psalms of Lauds (in fact for a time they
where Lauds which was tagged on the end of Vigils). The
only breviary that keeps them intact and together is the original Benedictine monastic breviary which is still prayed in many places, either in reformed post-Vatican II form (Solesmes, St. Wandrille in France to name a couple) or in pre-Vatican II form (Barroux, Fontgonmbault, in France, Clear Creek in the USA).
The Piux X psalter also broke up psalms in to more or less equal lengths (something that is often criticized with the 4-week LOTH but is by no means an innovation of that breviary), and that required ditching traditional antiphons and composing new ones with no real grounding in tradition. It caused a major hurricane to rip through the antiphonary.
Most of the reforms, in fact, that made the 4-week LOTH possible were in fact instituted by Pius X, and his reforms were no less controversial then, than the LOTH was 60 years later.
I find it tenuous at best to call the 1960 breviary part of a “millennia-old tradition”. IMHO it really has too many novelties to allow it to carry that attribute. The traditional Benedictine Office is the only one still in use that can make that claim, and moreover, it has the advantage of having been adapted to the post-Vatican II liturgical years and prayers. So you can use an adapted version with the 1970 Missal, or you can use the pre-Vatican II version with the 1962 Missal. It provides a MUCH greater continuity with a millennial tradition, as it dates back to the mid-500s, and many other breviaries grew from it.
For me the 1960 Breviary is a short anomaly using a 50-year transitional psalter; 50 years is very short in Church life. It started the trend of abbreviating the psalter for secular use; for example 9 instead of 12 psalms at Matins (the Benedictine Office is still 12 psalms).
That the reforms eventually led to a 4-week cycle for secular use, also made it significantly within greater reach of the laity, and that in itself is a good thing. I would much rather use a 4-week cycle that exposes me to all the richness of the entire psalter, than being only able to pray a portion of the 1960 (or Benedictine) Breviary which forces me to miss out on many beautiful psalms.
That’s not to say that there aren’t any shortcomings with the 1970 LOTH, there are: removal of imprecatory psalms and verses, a NT canticle at Vespers that chants awkwardly, and again the need to come up with some new antiphons (fortunately that has been corrected thanks to Solesmes and the Communité St-Martin, both of which have released excellent books allowing one to chant the LOTH in Gregorian chant, with new antiphons on traditional melodies), to name a few.
FWIW, I use a one-week post-Vatican II Benedictine psalter (Schema B), the same one used at our abbey. And the abbey (and I) still do it in Latin Gregorian chant for Lauds and Vespers. Like the traditional Benedictine Office, and unlike the 1960 Breviary, it does not arbitrarily break up psalms other than Psalm 118’s natural divisions since it is so long. Moreover they are distributed such that the psalm suits the character of the hour (St. Benedict didn’t pay as much heed to that, instead often preferring to keep them together in numerical order, though some were set aside as specific for Lauds, Vespers and Compline).