Structurally, "The Slave Traders" is a real refinement of the concept that Spooner introduced with "A Land of Fear" (the opening episode of 'The Reign of Terror'). In much the same way that his previous story traded on the tropes of the French Revolution, this story plays with the pop culture version of Rome more than the actual history. That isn't to say there's no overlap between the two, but Spooner has left historical Rome behind for schoolboy adventure stories, sword-and-sandal epics, and of course William Shakespeare (who gets a call-out in this episode thanks to Ian).
In fact, it's Shakespeare who probably proves to be the strongest influence on this story--the idioms of speech are different, but this could easily pass for a Shakespearean comedy. The slave traders are written in the style of his villains, rough and lacking in virtue but not significant enough to be more than incidental players in a grander tale. The confusion of identity between the recently-deceased Maximus Pettulian and the alive and utterly mischievous Doctor is straight out of any number of plays (although 'As You Like It' pops immediately to mind, disguises and mistaken identity is a hallmark of Shakespeare's comedic works). The splitting up of fast friends, the complications and peril that owe as much to contrivance as to character...it's all very much reminiscent of the Bard, although his Roman plays rarely had humor in their hearts.
For all that Shakespeare is taught in English class, though, this is light-years from Doctor Who's original mandate as an educational program. This is a story that could never have happened without Spooner's first Doctor Who script breaking that new ground and giving us a non-didactic (and not necessarily accurate) vision of history...but while 'The Reign of Terror' had trouble filling its running time with interest once three quarters of the cast were trapped in Conciergerie Prison and the fourth was stuck miles outside of Paris, here Spooner has figured out how to use his Roman tropes as a hook to hang a ton of plot points on. The Doctor and Vicki are jumping head-first into palace intrigue (and note that wonderfully, their reason is simply that it sounds like fun--we're light-years away from 'Marco Polo' or even 'Reign', where the goal was simply to return to the TARDIS as soon as possible) and Ian and Barbara's capture and subsequent separation allows them to explore their own individual plotlines. Spooner has learned his lesson; a plot like this has to feel overstuffed with incident to work.
It also needs great character work from the regulars, and we get it. Hartnell is now fully occupying the center of the narrative as the Doctor--his dialogue with Ian about pipes could have come out of the mouth of Matt Smith with very little alteration. Ian and Barbara have adjusted to their new, fundamentally reactive jobs of feeding the Doctor straight lines...but Vicki is where the series has really upgraded from one Spooner script to another. This isn't to blame Carole Ann Ford for the production team's decision to write her character as a shrieking neurotic, but Susan absolutely needed to leave the show in order for it to get to this point. Vicki is excited to be adventuring with the Doctor. She's bored with the placid villa life and actively looking for trouble--it's worth noticing that her efforts to dissuade the Doctor from impersonating Pettulian are almost entirely pragmatic, not ethical. She's behaving like the Doctor's partner in crime rather than his petrified granddaughter, and it really allows the show more freedom to become an adventure story rather than a history lesson. Like "A Land of Fear" before it, "The Slave Traders" is a huge step forward towards our modern conception of Doctor Who.