I have a question and I ask this with all sincerity.
Why should I become a Byzantine Catholic instead of
becoming Orthodox?
I am still too new to posting to be able to view profiles, but most responders here note that you are currently Latin Rite.
I do think a lot boils down to whether you are willing to give up being in union with the Pope. If you transfer rites within the
sui iuris churches, you will still be in union with Rome. You could become a parishioner of an Eastern Catholic church without transferring -- but if marriage, sacraments for children, future vocation to diaconate or priesthood are considerations, you'll have to transfer. If you become Orthodox, you will be rejecting Catholicism.
I have an aunt & uncle who have been in many places, church-wise. My aunt (Mom's mom, my own godmother) was raised Methodist, but my late maternal grandfather was born/raised Ruthenian Byzantine. He fell away from Catholicism altogether, and had questions about Faith altogether, so when he and my late grandmother were married, they went to her church, which was Methodist. My uncle, Aunt K's husband, was raised Roman Catholic, but stopped practicing at some point in time. My aunt has been a part of many different churches over the course of many years. Within the past 15 years, they came into the Roman Catholic church, were attending an indult Tridentine Mass sometimes, regular novus ordo RC Masses at other times. At some point my aunt became upset at her priest or the parish or something, and led the charge on them converting (Not transferring to an EC Rite) to Russian Orthodox.
When they converted, they had to LITERALLY "turn their backs" and renounce the Catholic Church as part of their conversion. If you've ever seen a Byzantine baptism, everyone turns their backs on evil at the start of the baptism. They face the back of church, and then turn forward to symbolize rejecting Satan and facing Christ. My aunt and uncle had to do this, but with regards to the Catholic Church. It was a stab to the heart when I learned they had done this. Even now I tear up, thinking of how, in a Catholic --> Orthodox coversion, the Catholic Church is basically viewed in the same way that Satan is in baptism.

Well, later, my aunt became upset about something in the Orthodox church (I think it was fasting regulations) and decided they were returning to the Catholic Church. Now they are attending a RC church again. Aunt K. is mad about something again, so who knows what will happen. I just pray for her, because she is clearly seeking something and either hasn't found it, or has found it and is either blind to it or is being obstinate and keeps running from what she has found. My uncle, on the other hand, embraced the chance to confess leaving the church and returning. I don't think he's going anywhere this time.
This is not in any way to tell you that one is better than the other. I'm just telling you to give you the idea of what you do if you leave the Catholic Faith and convert to the Orthodox Faith. This was at least the case in a church in the southern tier of NY.
Now, if you are RC and are feeling pulled to something else, and are finding it in Eastern Christianity, I can relate on that, too. Around 2000 we began our own journey toward the East. We had moved from southwestern PA, where the churches were more traditional and reverent, and orthodox (little 'o'), where we had met at a Catholic college, to central NY. The RC Diocese of Rochester, NY was a culture shock to us, where Masses in most churches really violated a lot of guidelines. Priests violated guidelines, and even the Bishop publicly stated that he disagreed with "no ordination for women," but would not disobey it. My husband & I went to a talk for men who felt they might be called to the diaconate, and the vocations director there stated to a woman who was there with her husband, that who knew? Women being ordained to the diaconate was not outside the realm of possibility. We walked out of Mass at the church closest to us on more than one occasion for the things they allowed, including the week that the resident nun stood up and delivered the homily.
Before anyone jumps on me for running away from problems in the RC and telling me we should have looked harder, well, we DID look harder. We drove a half hour to the least offensive parish in the area and were very involved there from 1999 - 2002. We drove 1-1/2 - 1-3/4 hours a couple times to attend an indult Mass but found we didn't feel at home there at all. But at the same time, we had started attending liturgy at the nearest Ruthenian Byzantine parish to us (1h. 15m. away) in about 2000 or 2001. We had run into one of our priest friends/former professors from college, who was going for bi-ritual faculties. He was so in love with Eastern Catholicism that he recommended that we seek out an EC church just to experience the Liturgy. We did, and instantly felt how he did. So we did our best to go to both churches for a while. Saturday at one church, Sunday at the other. Then in 2002, we found a Ukrainian-Byzantine church much closer to us. We went to Liturgy there and instantly hit it off with the pastor. It was a very small church, and not very active, but it was close and the pastor was very excited about helping us be involved there and helping if we would decide to seek a Transfer of Rite.
In 2002 our pastor at the RC church was being transferred to a church in NYC. (he was an ordered priest, not a diocesan priest, so went wherever his Provincial told him, which could be outside of the diocese) We felt it was a good time for us to really investigate whether we were truly being drawn to the East or not, so that October we joined the little Ukrainian parish and attended regularly. From Oct. 2002 - Oct. 2003 we prayed, and remained involved, and prayed some more. All the time we felt more and more drawn to the East. As I said, I still had relatives on my mom's side who were Byzantine Catholic. There are Byzantine Catholics on my dad's side as well. My paternal grandmother's home parish is a Ruthenian-Byzantine one. Her brother, my godfather, is still an active member there. I was at my cousins' weddings there.
The more we went, it hit me how Byzantine spirituality was so much more like my own. We felt we had found home, despite it really being a dying parish. It may have started with us simply seeing a more orthodox church than what was in the RC diocese. But it grew into much, much more. I knew that I wanted our son (already baptized) and any future children to receive their sacraments in the Byzantine Church. By 2003 we were finally expecting our second child, and it was very important to us for this child to be baptized Eastern Catholic.
In Oct. 2003, we wrote our letters to apply for a Canonical Transfer of Rite from the RC Church to the Ukrainian-Byzantine church. In Jan. 2004 the transfer was approved. We have "officially" been Byzantine Catholics for seven years. Our daughter was born in Feb. 2004, and was baptized, chrismated & communed in March 2004. Our son had been baptized in the RC church in '96, but had his First Confession, First Holy Communion and Chrismation in March 2004. While there are certain times of year that we keep some traditions that are of "both lungs," and when visiting family (still RC) we make it to RC Mass more than Byz.Cath. Divine Liturgy, I have never once looked back and felt that we made a mistake in transferring. Not once.
Over the summer we moved to my dad's home area because of my husband's job. In October, we became parishioners at my grandmother's home parish, and love it. We are home, you know?
Before you make any decisions, have you started "checking out" Eastern churches? It is best to become an active parishioner at an Eastern church and spend at least a year prayerfully discerning whether you are feeling called to change Rites, before applying for the transfer. There is nothing wrong with attending Orthodox Liturgy as well, without receiving Communion at this time. You'll find that the two are very similar in Liturgy and tradition.
But as a previous poster said, the decision to leave the Catholic Faith should not be taken lightly. Once you leave, you will no longer be able to receive Communion in a Catholic church. A Greek Orthodox friend of mine told me that can be cause for excommunication from the Orthodox church. So if you leave, you won't be able to receive Communion in your family's church if you're ever there for a Mass. Just some things to think about.
I'm sorry for the very lengthy reply. Only you, guided by the Holy Spirit, can decide what to do. There are many similarities between Orthodox and Eastern Catholic, so I can see why you're wondering which would be better for you. There are some theological differences between the two, but a huge difference is that with one you will still be in union with Rome, and with the other, you will not.
Many prayers for you as you discern what to do!
God Bless.