Why is it that different economic sectors are growing at different speeds?

At their core, economic sectors are just work that people do. When the work that needs done changes, economic sectors shift accordingly. One of the central benefits of a free market economy is that it allows industries to be created and destroyed as the work that needs done changes, a process called creative destruction.

Probably the biggest cause of shifts in economic sectors is technology.

As new technologies are invented, they make old technologies obsolete, and demand goes up for people who can use the new technologies and down for people who can use the old technologies.

For example: There aren't a lot of people employed as computers anymore! That used to be a profession, in case you didn't know; it was mainly done by young women, who used slide rules to perform and check calculations for statisticians and engineers. Once microchip technology allowed us to build the machine computers that were faster and more accurate (and today, millions of times faster!), people who were employed as the profession computers lost their jobs and had to do something else, while suddenly there is a huge new market for software programmers and web developers.

Technology can also radically alter productivity in a given industry, so that far less labor is needed to achieve the same output, thus reducing labor demand in that industry. Agriculture used to be basically the only job, done by about 95% of the population; but then we invented fertilizer, irrigation, industrial farming equipment, etc., so that now one farmer can produce as much as a hundred farmers did a thousand years ago. As a result, we produce more food than ever---even per person, despite a rapidly growing population---despite having only about 1% of our population engaged in farming.

Related to changes in technology and productivity, there is also a general pattern we tend to observe as a country becomes more economically developed; first, people shift out of farming into labor-intensive manufacturing, then they get into more capital-intensive manufacturing, and finally they start moving over to services, ranging from restaurants to healthcare to banking. First World countries usually have a huge service sector, typically dominated by finance and healthcare, while the poorest Third World countries are almost all agriculture and moderately-developed countries have manufacturing as their largest sector.