Murder in the Cathedral is a drama written as verse. In T.S. Eliot's time, major dramas had not been written in verse for around 300 years. For example, in 1671 John Milton published Samson Agonistes. It was written as a verse drama but Milton specified in the preface that it was not to be performed. So one way in which Murder in the Cathedral is different from Hamlet, Dr. Faustus, The Alchemist, etc. is that, though they are all verse dramas, Eliot's is a revival of an unpopular and disused form while the others were the favored and popular modes of drama in their eras. They are similar, though, in that each drew on other sources, some drew on history, like Shakespeare's history dramas, and some on legend or tale like Dr. Faustus.
Though done in verse, Murder in the Cathedral and the others are different in that the verse in the early dramas is written using the vernacular of the era and Eliot's is written using the vernacular of his era. This sounds like a similarity, but it is a similarity that leads to a great difference in style, tone and musicality of the verse.
The thematic concerns are somewhat different in that while all discuss important weighty themes, Eliot turns his two themes of the spirit versus the flesh and obedience into vehicles that illustrate then contemporary issues such as privacy and religious intrusion into individuals' lives.
The tragic hero of Murder in the Cathedral, Thomas Becket, differs from Aristotelian and Elizabethan tragic heroes in that Eliot leaves him free of a "tragic flaw." He has no internal weakness or failure of character that blinds him or misleads him, he commits no tragic error in judgement; he has no ambitions that drive him into vain misdeeds, etc. In addition, Becket's death sets a new heroic style when his death is one that fills the audience with the idea of peace and hope instead of pity and agony. Eliot also divides the physical tragedy of Becket's murder from the spiritual reality of Becket's life, which is that his spiritual life supersedes his physical death.
My first thought in response to this question lies in the fact that Murder in the Cathedral is based in historical fact. Eliot drew his information from the first person accounts of Clerk Edward Grim who witnessed the event and write about it himself. The other plays, while they draw heavily on universal themes, such as the revenge theme in Hamlet, are fictional. They could happen, to be certain, but the murder of Archbishop Thomas Beckett did, in fact, happen. Eliot's work, therefore, could be defined as historical fiction. There is a degree of dramatic license in the play, but the events themselves and the details surrounding the main event are meant to be as accurate as possible.