I've always thought that the words parishioner and congregant meant the same thing and could be used interchangeably within the context of someone who attends a place of worship. Are there any differences in meaning between these two words or appropriate usage for each?
Parishioner and congregant refer to members of a particular local faith community. The requirements for membership, of course, vary considerably, but for the most part, simply attending services at a church does not make one a parishioner or congregant of that church any more than visiting a country makes one a citizen of it.
Sectarian considerations govern which is the more appropriate term.
Parishioner is older by a good measure. A parish is an ecclesiastical territory, a section of an episcopal see (e.g. a diocese or archdiocese). Traditionally, any inhabitant of that territory would have been expected to attend services at the local parish church, and all would have been parishioners. The Catholic, Anglican, and Orthodox churches remain organized in this manner, but the term "parish" is used even by some denominational bodies without episcopal administration, so making the members parishioners.
Congregant is broader, in that it refers to the regular members of any local congregation. That local congregation may be a parish, but it might also be a local church or meeting house of a tradition that does not use the term parish, such as the Baptists or Mormons— or for that matter, Muslims or Jews.
To refer more generally to those attending services at a particular time, you could simply say worshippers or attendees; for all adherents, there are a variety of terms employed, such as the brethren or the faithful, or the more mundane churchgoers or the observant; communicants captures the sense of those in communion with the Church as opposed to outsiders.
In the US and US state territory under Common Law, parishoners are associated with residency within extra-governmental, quasi-jurisdictional, territorial divisions, called parishes--these are separate and apart from, and subordinate to, state territorial political divisions and instrumentalities--under jurisdiction of an heirarchically structured religious institution, most notably, the Roman Catholic Church headquartered in Rome, but also of derivative and offshoot organizations--but, as established under Amendment 1, the US Constitution, only insofar as any such resident is or voluntarily becomes a pledged or "customary" adherent (a congregant) subject to the aims, teachings, doctrines, authority, and jurisdiction of said parish'es congregation, both clergy and laity, as represented, headquartered, housed and served by a permanent church, Cathredal, mission, temple, ....
Tracing from feudal times when Church and State vied for primacy, demarcation of parish boundaries today are of a generally more ill-defined, more fluid, more ad hoc nature. In the abstract, it could be said that spatial density distribution of church member and attendee residency as between one parish and a neighboring parish in large measure defines interparish boundaries, with one exception. Under Romanic law in Lousiana, parish and political subdivisions coincide: parishes (for example, the Parish of New Orleans) take the place of county and county subdivisions. (Incidentally, the best known distinction as between Common and Romanic Law, respectfully, it the presumption of innocence and of guilt.)
Except for association with a territory, the word, congregant, subsumes and is readily exchangeable with the ascription, parishioner. An active parishioner is a congregant. A congregant is not necessarily a parishioner.
I am the member of a two point parish--in that, two separate church congregations share one pastor and all of the costs associated with our pastor--and that pastor shares his/her time between the two congregations. Therefore, I am the congregant of my individual church; however, I am also a parishioner of the total parish.
Sometimes interchangeable, but not always. A member of a Catholic church or Jewish synagogue, for example, could be congregants (they congregate, or gather) but not parishioners, as there is no parish, no pastor. Parishioner usually connotates a Protestant, which we have many of in the USA.