Is the Catholic understanding significantly different from the Orthodox one? All my research has been into Orthodoxy, so I'm a bit backwards and always assuming that the Catholics are like the Orthodox instead of that the Orthodox are like the Catholics.
No - the Orthodox Church and the Catholic Church are two lungs, an eastern one and a western one, of the same Church, and the Church is in schism between its two halves over very specific structural issues.
In terms of the fundamental beliefs - they are expressed in the Nicene Creed. Orthodox/Catholics believe in each and every word of the Creed - those are the fundamental and indisputable elements of the religion.
Which is why the primary theological division between the two was the matter of the procession of the Holy Spirit - from Father alone, or from Father and Son both? That was the theological basis for the great schism, and it remains the most important point of difference.
The rest of the difference arises from ecclesiology, not in general, bust specifically related to the role of Peter and his successors in the clergy. Specifically, is the heir of Peter's See, the Pope in Rome, a leader primus inter pares, senior in respect, or is he senior in both respect and authority.
One can find all sorts of differences in praxis all across the Church - Mexican Catholicism looks and feels very different from Swedish Catholicism -and one can find Catholic and Orthodox churches that are difficult to distinguish (Syrian Orthodox and Syriac Catholics, or Greek Orthodox and Byzantine Rite Catholics, for example).
Essentially, all of the fundamental essential doctrinal points of Catholicism that are opposed by Protestantism are also fundamental doctrines of the Orthodox Church. Politically, Protestants generally find Orthodoxy much more congenial, because Orthodoxy does not have the Papacy (though it does have authoritative patriarchs within each national church, each of whom is pope), and because there is no the long history of direct conflict and warfare between Protestantism and Orthodoxy as there is between Protestantism and Catholicism.
For the most part, Protestants who are firm in their Protestantism find Orthodoxy charming at first blush, and then find it to be wrong in all of the Catholic ways.
For the most part, Catholics find Orthodoxy to be the same religion as Catholicism and have little to quarrel about.
For the most part, the Orthodox find Catholicism to be their closest kin, but are keenly aware of each of the points of divergence.
For the most part, very senior Catholic and Greek Orthodox clergy find the differences to be stylistic and more of appearance than substance, but still too difficult to completely bridge at the present time.
For example: Catholics will say that Orthodoxy and Catholicism are essentially the same Church.
The Orthodox will disagree with that.
Protestants tend to be much more sympathetic to the identical doctrines when they come from Orthodox mouths than Catholic ones.