For example, the capital of New York is Albany, not NYC. The capital of Illinois is Springfield, not Chicago. The capital of California is Sacramento, not LA. The capital of Florida is Tallahassee, not Miami. The capital of Louisiana is Baton Rouge, not New Orleans. The capital of Maryland is Annapolis, not Baltimore. The capital of Washington is Olympia, not Seattle. Looking at the list of state capitals on Wiki, it seems some states - Arizona, Georgia, Massachusetts - do buck this trend, but most state capitals seem not to be the largest cities in the state.
Is it just historical - were these places more significant when they were picked as capital? Or was there some policy of trying to separate the bases of economic and political power in their states?
Some were more significant at that point in time, than they are today. Some was also politics.
In CA for example: Monterey was the original capital, but the legislature met in San Jose until 1851, Vallejo in 1852, and Benicia in 1853.
Sacramento was the first incorporated city in California. It was only ninety miles inland from San Francisco, and because of the '49 gold strike it was a major distribution center and center of gold mining activities. It was the end-point of the First Transcontinental Railroad, and most of the businesses that came west, set down there. It was inland, and so was harder to invade from the sea, but on the river to a major port, so it could still do business from sea-borne trade. It was also the only foreigner friendly area in the former Mexican lands. It had the first fire protection company. Etc, etc.
Between January 1848 and December 1849, the population of San Francisco (Yerba Buena) increased from 1,000 to 25,000 because of the gold rush - and a lot of those people were in transit to Sacramento. It didn't really start to expand until 1860-1880.
Los Angeles didn't overtake SF in population until 1920. LA was small Mexican town (started with 11 families) that changed little in the three decades after 1848. It didn't start to grow until the Santa Fe railroad line from Chicago completed in 1885.
Arizona used to be below New Mexico, and first capital was Tucson, especially if you were Confederate. Feds redrew the maps more into what we know today, and Prescott was their choice. Tucson was the most developed at the time, but had Confederate sympathies, so they moved back to Prescott (which was cooler). Eventually they decided to split the difference, and picked a point in-between the two. So in AZ's case it wasn't all about the biggest, or best developed economically - it was a political move.
With 300+ years of history (for some states), things change.
I can speak to Vermont, where the State capital is very small - if I recall correctly, it is the smallest State capital in the US. Essentially, it is due to geography.
Montpelier is fairly central, an important consideration in the late 1700s. It is on a river (the Winooski) that bisects the Green Mountains that run more or less down the center of the state. (Fun but not really relevant fact: Vermont used to have an unofficial "Mountain Rule" whereby the Republican Party would choose a candidate from east of the mountains, then one from West of the mountains. Back when the Republican Party completely dominated state politics - post Civil War to the 1960s - this meant each successive Governor was from one side then the other.)
State capitals were chosen by state legislatures, and are often centrally located so that all representatives have equal travel time from the capital to their districts. The most economically important city is the state will become so over time based on trade and industry.
Maryland is a good example of this. Annapolis was centrally located by travel time (by boat) when it became the capital. Baltimore did not exist when Annapolis became the capital, but it grew and prospered to become the economic center because of its port and railroads.
I can answer why Tallahassee is the capital of Florida. St. Augustine on the East Coast and Pensacola on the Gulf Coast were basically the only two significant cities in Florida in the 1820's, soon after the U.S. acquired and consolidated Florida Territory. Territorial legislators tried to meet in session in alternating cities, but the travel from one city to the other took nearly three weeks for the opposing city to travel.
The land around Tallahassee (modern day Leon County) was roughly equidistant between the two cities, and was so chosen.
Obviously, the state grew significantly over the next 200 years, and the bulk of the state is in the Central and Southern parts, and no longer the North. North Florida today retains much of its' Southern culture while the Central and Southern parts comprise a mixed bag of immigrants and transplants from around the US and the world.
Source: http://dos.myflorida.com/florida-facts/florida-history/the-capitol/
But, to correct the record on one thing if I may: Miami is not the largest city in Florida, Jacksonville is.
Texas is interesting in that during the Republic, eight different cities had been the capital. As a state, it's only been Austin.
Majority of the eight capitals of the Republic of Texas are east, but the second president, Mirabeau Lamar, felt it was best to have a capital closer to the Pacific (he thought Texas would extend to the Pacific. He felt a buffalo hunting range near the Colorado River would be better (this hunting site is now downtown Austin).
He also hated Sam Houston and hated the thought of the capital being in Houston. At the time, there were contracts that the capital would be a city for x-amount of years (each contract had a different number). Lamar pressured the legislature to break the contract for Houston and create a commission to site the capital further west. They settled for the city of Austin (an imaginary town named after the recently deceased Stephen F. Austin).
There was no trade routes and it was vulnerable to Mexican and Native American attacks. When Sam Houston became president again, he attempted moved the archives to Houston for protection. He knew if the archives were moved, the government would have to follow. The small population of Austin (excluding government officials the city was basically just blacksmiths and innkeepers) fought back and caused troops sent by Houston to retreat. The capital did move to Houston afterward(then Washington-on-the-Brazos for the second time), but Austin survived.
When Anson Jones became president in 1845, the capital was moved back to Austin. The city was to be capital for a few years and then the issue would be reconsidered. In late 1845, Texas entered the union. When time came to decide if the capital would stay in Texas, I suspect everyone was tired of moving the capital and Austin became the permanent capital. People are surprised that Austin is the fourth largest city in the state (and second most populous state capital behind Phoenix), but in 1850 (a few years after annexation), Austin was the 16th most populous city in Texas.
In Kentucky, small Frankfort was chosen because Lexington and Lousiville (the state's two large cities) were unacceptable to the other. Frankfort lies roughly between them.