Looking at Europe, most countries' largest cities are also their capitals.
But with the United States, often the largest and most well known cities are not their respective states' capital. I think most Europeans like me were rather surprised when they first learned that NYC is not in fact the capital of New York State, and Los Angeles is not the capital of California, and they've never heard of those states' actual capitals Albany and Sacramento.
And looking at other American States, that seems to be a rather common thing: Seattle is not the capital of Washington, Miami is not the capital of Florida, Portland not the capital of Oregon, Houston not the capital of Texas and so on. There are exceptions of course, like Nashville in Tennessee or Providence in Rhode Island. But for the most part I think the first pattern applies: largest city ≠ state capital.
Is there an explanation for this?
There's number of reasons for this to be so.
(1) Population Growth -
America's population has grown very quickly by historical standards. The main centers of early population did not always grow into the largest cities due to geographical constraints or changing economic conditions. For example, Indiana had as territorial capitols Corydon and Vincennes (both tiny towns on the Ohio and Wabash rivers, respectively) before moving the capitol to Indianapolis (at the time a small and disease ridden planned city).
One example of a capitol city that time has slipped by is Frankfort, KY. It's really not all that far from Louisville, the major city in Kentucky, but is among the smallest state capitols by population.
(2) Comparative Past Size
The explosion of mega-cities, such as LA, Chicago, and even New York is also a more modern phenomena. While many of these cities were also large in the past, the difference in size between their population and that of the rest of the state is different. This meant not only a greater parity in political power between the main city and everywhere else, but also that the balance of power lay with everyone else. They would not be too keen on the big city also getting all the political power.
One example of this would be Albany, New York.
(3) Geography
Things are best administered from a relatively central location. The choice of state capitols will typically reflect a concern for geographic centrality: Sacramento, Salem, Indianapolis, Nashville, Atlanta, Jackson, Pierre, Bismarck, Madison, Springfield, Columbus, Ann Arbor, etc.
Depends on the state, but for New York and Albany, this old answer from /u/discovering_nyc might be of interest for you.