The term “Traditional Catholicism” is usually used to mean Western Catholicism as it was prior to the 1960’s, whether in a reconstructionist sense or simply in the sense of preference those aspects of Western Catholic practice which have been preserved, at least as options, from that time.
Eastern Catholicism on the other hand is its own tradition, or rather several distinct traditions. Thus it is most proper to compare Eastern Catholicism to Western (aka “Latin rite” or simply “Latin”) Catholicism as a whole rather than with any sub-category some Latins may choose to make for themselves.
Fundamentally the difference arose mostly due to geographical separation over many centuries. The Church began in Jerusalem, geographically in the East but culturally in the Hebrew ecclesial culture which essentially did not continue into future centuries. From this Jewish starting point missionaries brought Christianity throughout the Greco-Roman and Indo-Iranian worlds, and beyond into places like Ethiopia. All of these far-flung communities were recipients of Apostolic Tradition, but probably thought and acted a little differently from the start. These differences tended to only increase as time went by, though of course there was plenty of cultural and intellectual exchange between different regions too. In some places a variety of factors caused some traditions to be mostly absorbed into another, such as the Gallican and insular Celtic traditions being largely absorbed into the Latin one (though leaving a mark on general Latin Catholicism; Gregorian Chant for instance probably bears strong traces of Gallican chant.)
Ultimately a situation has developed in which about 98% of Catholics (I suspect a somewhat smaller percentage of practicing Catholics who value orthodoxy, though I can’t prove that) belong to the Latin tradition that originated in Western and Central Europe and has since spread throughout the world. The rest of the Catholic Church is made up of other traditions, generally in their own self-governing Churches which are in communion with the Pope and therefore with the rest of the Catholic Church. Because these traditions originated east of Western and Central Europe, they are called the Eastern Churches or Eastern Traditions. Collectively they make Eastern Catholicism, though it must be emphasized that they consist of several different traditions. For instance an Armenian Catholic may or may not feel he has more in common with a Ukrainian Catholic than with a Latin Catholic.
It is tempting to caricaturize the different traditions, but this generally leads to misleading conclusions. For example it is often said that the East is more mystical and the West is more rational, but this ignores the powerful Western mystical tradition and the learned Eastern intellectual tradition. Also such comparisons can lead to misunderstandings and hurt feelings. For instance both Easterners and Westerners sometimes claim to be the tradition that puts greater focus on the incarnation. In fact any tradition that put less focus on the incarnation would be less Christian than the other, and so we should not imply this by claiming to be more “incarnational” than any other Catholic tradition.
As for where an individual Catholic stands, I personally would caution against changing from one tradition to another based on personally liking the other. Both, or all rather, of these ecclesial traditions are very deep and very culturally rooted. In general, I would uphold the principle that a person should stay within the tradition from which they received the Gospel. In the case of a Protestant (a member of a separated but still Western tradition) who is converted to Catholicism through contact with the Latin rite Church, the Latin rite Church is probably the home for him even if he grows into a great appreciation and love for the East and is influenced by it in his private spirituality. Same thing goes with an Eastern Catholic who grows to love Western art, the Rosary, etc.; the East will probably still be his best home.
