I suspect the difference in usage is related to the difference in how usage of the various fruits developed in different parts of the world.
Specifically, these plants are native to the Americas, where they were in cultivation long before Columbus arrived. They thus would have been first encountered by English-speakers in the Americas as a "going concern". The various fruits would likely have been a welcome and even necessary part of European Americans' (in the continental sense) culinary repertoire.
Names for such common plants and foods would have been practical, and might have arisen from indigenous terminology (chili); analogy with known, old world ingredients (pepper); or descriptive terms (a "pepper" that is bell-shaped = bell-pepper).
In other parts of the world, on the other hand, the plants would have first been known as exotic cultivars, and might have made their way into ordinary cultivation and kitchens by way of botanists, who would naturally be more concerned with precise terminology. Places introduced to the plants after this naming was established would be more likely to use it.
The OED's evidence supports the hypothesis that pepper is the older English term for the fruits of genus Capsicum. The OED attests this usage to 1578:
1578 H. Lyte tr. R. Dodoens Niewe Herball v. lxix. 634 The Indian
Pepper [Du. Peper van Indien, Fr. Poyure d'Inde] hath square stalkes.
1693 Philos. Trans. (Royal Soc.) 17 621 A long Pepper from Brazil.
1707 H. Sloane Voy. Islands I. 241 Bell Pepper. The fruit is
large..somewhat shaped like a bell.
. . .
("pepper, n." OED Online. Oxford University Press, June 2016. Web. 24 August 2016. Sense 3.)
Capsicum, on the other hand, is first attested as a botanical term for the plant in 1664, and as a term for the plant's fruit in 1725. ("capsicum, n." OED Online. Oxford University Press, June 2016. Web. 24 August 2016.)
Thus I think the sequence of events might have been:
- Colonists in the Americas are introduced to plants that they can use in place of the (much harder-to-come-by) Old World spice called pepper.
- Being of a practical, and perhaps somewhat nostalgic, bent, and therefore unwilling to coin new words when perfectly serviceable old ones exist, the colonists called these "peppery" fruits peppers, with identifying adjectives when necessary.
- These wonderful plants were sent by colonists back to their parent countries, where they were adopted with some enthusiasm, often under the common names used by colonists.
- At approximately the same time as 3, Old World botanists acquired specimens of these plants, and classified and named them according to conventions of the time.
- Folks who had been using these plants for decades or millennia continued to use whatever terms they'd been using all along outside of the scientific community.
- Eventually the plants made their way to cooks and farmers in other parts of the world, now stamped with "official" names, which became the "common" names. This might have been especially true in places where the original plants that had inspired the analogous name were cultivated.
If this is correct, I would expect the term Capsicum to be least familiar in the Americas, most familiar in non-American colonies and non-colonial parts of the world, with perhaps some non-scientific usage in Europe. (I can confirm that it's very unfamiliar in the US, but not the rest.)