I'm studying Op. 22 by Anton Webern for my AS music. I have 2 teachers and a revision guide, each of which uses one of the terms listed in the question and avoids the other two. I just wanted to see if there is a distinction between the terms and if so what it is.
Serialism refers to a number of compositional techniques or styles, the most well-known of which is 12-note serialism. Generally, serialism means music which is governed by one or more series of values (often pitch, but sometimes dynamics, tempi, rhythm) which are reused throughout the piece to give a sense of unity.
In 12-note serialism, a specific ordering of the 12 notes of the chromatic scale called the tone row is defined and reused in various forms throughout the piece. The first appearance of this tone row is called the prime order. The tone row is usually repeated throughout the piece in different forms. It may appear again subsequently in different orders and inversions, (retrograde, inversion, retrograde inversion).
So, to answer the question, there isn't a difference as such between the terms you mention, just that the term 'prime' describes the particular iteration of the 12-note tone row. I have not come across the term "serial" describing a tone row before, I think it refers more to the actual 'serial process' of composition.
See the Wikipedia entries for serialism and tone rows. If you want some extra reading/listening, comapre Webern and Schoenberg's 12-note serialism to the broader interpretations of the post-war composers, where they used pitch, duration, dynamics, AND articulation as series.