Two different species, and the difference that matters to you is where they live. King penguins are sub-Antarctic birds: South Georgia has the enormous colonies, and they also breed in the Falklands. Emperors breed far to the south of the sites a normal cruise visits, and reaching a colony takes a specialised expedition.
If somebody shows you a photograph of a vast colony with thousands of orange-flashed penguins on a coastal plain, that is South Georgia and those are kings.
Emperors are the larger of the two. Kings are the second largest, and their colonies are what most people picture when they imagine a penguin city.
You will not see them on the same trip. Kings mean a South Georgia or Falklands itinerary. Emperors mean a Weddell Sea or Snow Hill operation with helicopters. Very few voyages attempt both.
Kings have a quirk that changes when you can go. Their breeding cycle runs longer than a year, so a colony holds birds at every stage at once. Adults courting, adults incubating, and big brown woolly chicks, all on the same beach on the same day. It means you can usually see king penguin chicks right through the South Georgia visitor season, which the Peninsula cannot offer for its own species.