What is the difference historically and dogmatically between the old catholic, traditional catholic, and the current catholic church?
First of all, we "traditional Catholics" simply ARE the current Roman Catholic Church, pure and simple. We are the protrusion in our own era of that Church which our Lord Jesus Christ founded all those centuries ago on the Rock of St. Peter.
There arises, however, much confusion from the fact that there are others who have attempted to steal our name, but they really have no right to it and in fact merely affix a name which does not describe them at all. The true theological definition of "Catholic" means that they hold the entire teaching of the Church, that they are meant for all sorts of people in all places, countries, ethnic backgrounds, races and other variations in physical appearance, and in all times from the founding of the Church until the end of time. We are, in fact, found in all nations of the world, though at present with our severely depleted numbers, some smaller countries may at present have only Catholic laity.
One such category of such is those known as "Old Catholics." They broke off from the Church back in the 1700's and 1800's, even holding an anti-Council in Munich to define their different beliefs from the beliefs of the real Catholic Church. They adopted many of the heretical beliefs of the Jansenists and the Gallicans, but maintain a valid succession of sacramental orders obtained illegally through one Bp. Dominic Varlet, and their biggest stronghold is in Utrecht, Holland. Some of their number, with perhaps more sympathy for authentic Catholic beliefs, have subtly renamed themselves "Old Roman Catholics," but their succession is derived from the rest of the Old Catholics and as such there is nothing actually Catholic about them.
Another of such category is those of the "Conciliar Church" (technically not actually a "church" per se, but rather a secular power with a false state religion). They broke off from the Church back in Vatican II when they redefined their formerly Catholic organization as being something other than the Church in a manner that permitted, for some season at least, some Catholics to continue as members of it while not adhering to their new state religion. It is this group that causes the greatest confusion in our times since many of their buildings (which we Catholics built before Vatican II) still have the name "Catholic" on their placards, misleading many into thinking that they are still Catholic, even while the fact that they are under new management is plainly conspicuous. Their separation was rather unusual (but not unique) in that they took over nearly all the existing facilities and personnel of what had formerly comprised the Catholic Church. The vast size we Catholics once built up they have hijacked and stolen away. Over time, the Catholics have by and large made their departure (or were driven out) continuing in the fullness of what it means to be a Catholic. The Novus Ordo religion of the Conciliar Church (properly called "Conciliar" since it was officially inaugurated at a Council, namely that of Vatican II) has much in common with the religion of the Old Catholics, and many of the same liturgical aberrations have turned up in both.
There also exists (though not mentioned in your question) those who call themselves "Anglo Catholics" by which they refer to the very highest "High Church" of the Anglican (Church of England, or Episcopalian) communion, namely that which within that schismatic body most closely resembles the Catholic Church. They too have their own subtler liturgical aberrations owing to having to at least nod in the direction of their "Low Church" brethren whose services really are much more typical of Protestant services in general. Apart from any rare instances in which any Catholic or Old Catholic or East Orthodox clerics have defected to it they have no valid sacramental orders.
It is interesting and indeed significant that among all the false claimants to the Catholic name all have modified their liturgies. But of course this merely reflects the fact that the law of prayer must always echo the law of belief and vice versa, so their different beliefs are one and all commemorated in their different liturgies.
Traditional Catholics alone preserve the fullness of the original and authentic belief and liturgy of the Church as imparted by Christ.
I have written extensively on the survival of the real and authentic Roman Catholic Church through this most trying period caused by Vatican II, as can be found online at my site: