What is the difference between a Geranium and a Pelargonium?

While the genera Geranium and Pelargonium are related, both being members of the family Geraniaceae, they are quite distinct. True geraniums, also known as cranesbills, referring to the shape of the fruit, for the most part have symmetrical flowers with ten fertile stamens. Most Pelargonium have bilaterally symmetrical flowers with up to seven of the ten stamens fertile. True Geraniums have a different seed dispersal technique than Pelargoniums. Geraniums fling their seeds away while Pelargonium seeds float away on the breeze and usually have a 'feathered ' end that Geraniums don't have. Of course, you can only see this when they are producing seeds.
Pelargoniums are tender perennials and occur naturally almost entirely within South Africa. Leave of true geraniums are usually deeply divided and cut while those of most groups of pelargoniums are not. Pelargoniums also have rather thick, succulent stems, originating as they do from areas where they have to withstand summer drought, whereas geraniums have the appearance of 'normal' herbaceous perennial plants, a mounding form of many many slender stems arising from a central core, and fibrous roots.
Plants sold at garden centers that are labeled "Ivy Geranium", "Scented Geranium" and "Zonal Geranium" are actually three different species of pelargonium. Some names that true geraniums are commonly known by are "Hardy Geranium", "Geranium" and "Cranes Bill".