Agriculture.
Due to the islands' low lying location and highly saline soil, only 0.83% of the land is suitable for cultivating crops. As a result, only 1.9% of the labor force works in the agricultural sector, and the main agricultural products are turtles, and other aquaculture. The majority of agricultural products (meat, livestock, etc.) are imported from other countries (particularly the United States), and so only 0.3% of the Cayman Island's GDP comes from agriculture. This is significantly smaller than the United States' agricultural contribution to the U.S. GDP, which is a whopping 5.33 times more, at 1.6%. This is likely due to the presence of more ample and fertile land in the United States (16.96% arable land), and thus, a larger agricultural industry with a bigger impact on the GDP. Since there is very little food produced in the Cayman Islands and most of it must be imported, 57% of the Cayman Islands' GDP is dedicated towards importing agricultural and consumer goods, which form almost 90% of the islands' goods.
The US's agricultural sector also is able to produce a greater variety of goods (from dairy to corn and all in between), which lessens the US's need to rely upon foreign countries for consumer goods and allows it to be a crucial trading partner to the Cayman Islands. The Cayman Islands have 741 people working in the agricultural industry (or 1.9% of 39,000), while the US has almost 11 million, so the US has more human capital, land, and physical capital (# of tractors in the US definitely exceeds tractors in Cayman Islands ◕‿◕) that can be devoted to agriculture. Also, this is reflected in the imports and exports of each country-the Cayman Islands, with its limited production capability, only exports $15.2m worth of turtle products and other consumer goods, while the US is able to export $1.6 trillion worth of products. 9.2% of the US's exports are agricultural products, so agriculture is a significant commodity that the US can use to earn revenue. The Cayman Islands must spend 57% of their GDP importing goods, while the US spends only 16% (however, considering the differences in GDP, the US would still end up spending more). All in all, the Cayman Islands aren't that big on agriculture, and prefer to obtain their food from foreign countries (particularly the US). It seems like that is working out quite well for them, as the Caymanians apparently "enjoy a standard of living comparable to that of Switzerland" (CIA Factbook). The cost of living as a result of these imports seems to be slightly higher (1 liter of milk is $2.5)-but then again, there is no sales tax.
Industry.
The Cayman Islands industry is also not as large and advanced as the US's, since it lacks both human and physical capital (and land). In addition, it would be quite difficult to cram a highly developed manufacturing industry along with a tourism industry into 264 sq km of land (smokestacks don't exactly go well with pristine beaches).
The industries of the Cayman Islands are focused on construction, which ties into tourism and offshore financial services (building hotels and places to register companies) and furniture construction (mostly mahogany wood, to be exported). 19.1% of the 39,000 strong labor force works in these industries, so although the industry does not contribute much in terms of consumer good production, many households still participate in the industry. This percentage is actually not that far from the United States, where 20.3% of the labor force works in industry. The difference lies in the number of people (same'o, same'o) because the US has a greater labor force, but also in the types of industries that the US has: electronics, steel, petroleum, vehicles, telecommunications, food processing, consumer goods, mining, etc.
The industries of the Cayman Islands are focused on construction, which ties into tourism and offshore financial services (building hotels and places to register companies) and furniture construction (mostly mahogany wood, to be exported). 19.1% of the 39,000 strong labor force works in these industries, so although the industry does not contribute much in terms of consumer good production, many households still participate in the industry. This percentage is actually not that far from the United States, where 20.3% of the labor force works in industry. The difference lies in the number of people (same'o, same'o) because the US has a greater labor force, but also in the types of industries that the US has: electronics, steel, petroleum, vehicles, telecommunications, food processing, consumer goods, mining, etc.
Photo courtesy of Economicpopulist.org and myself
Given the wider variety of industries, the US's industry is much more developed and includes many more professions than the Cayman Islands' industries, which are limited in scope. Despite this, industry's percentage contribution to GDP in the Cayman Islands actually is greater than the US, at 28.4% of the GDP compared to the US's 20.7%. In both countries, industry is significant, but the higher number for the Cayman Islands is likely due to its smaller GDP, allowing industry's percentage to make a greater impact, though it isn't as productive as the US's. In terms of cash, the US's industry still produces more and contributes more to the GDP. So, the Cayman Islands' industry is not as advanced, nor varied, as the US's industry, but it still is an important source of jobs for households, and significantly contributes to GDP.
Services.
The services sector is definitely the Cayman Islands' most developed economic sector, and the major service sectors are tourism, banking and finance, and insurance. 78.1% of the labor force participates in the service sector, which is comparable to the US's 79.1%. The US's service sector is more developed compared to the Cayman Islands because it has more people, and contributes 77.7% of the GDP, whereas the Cayman Islands' service sector contributes 71.3% to its GDP. However, that does not mean that the Cayman Islands' service sector is insignificant-more than 93,000 banks are registered within the islands and over a million visitors (nearly half from the US) visit the Cayman Islands annually. The Cayman Islands' service sector is more focused than the US's, as the US has a very large variety of services for the 79.1% to work in (see above pie chart, left) while the Cayman Islands' major service sectors are offshore finances and tourism. That is not necessarily a bad thing, as the Caymanians seem to have enough jobs to go around (4.7% unemployment, not as high as the US's 6.2%-consider population difference) even with limited service sectors.