The premise of this question is valid mainly in comparison to Europe. In other parts of the world (e.g. Africa, southeast Asia), public transportation in the sense of subways, rail lines, and buses is generally much less developed than in comparably-sized US cities (with notable exceptions like Japan and Bangkok). As far as I can tell, there are a few factors driving this disparity:
1) Car-friendliness. The US is a very young country. As a result, many cities in the US, particularly in the western half of the country (think Texas, Colorado, California), were developed with cars in mind. Sure, Houston existed before cars, but it was small enough by the time cars were mass-marketed that roads and city planning could be adapted to them. The result has been wide roads, extensive freeways/expressways, and plenty of parking, encouraging driving.
European cities, by contrast, tend to exhibit very dense development, narrow roads, and limited parking. There's just no room to expand to accommodate cars. As a result, roads are often congested and parking is non-existent or extremely expensive once you get to your destination. Worse still for would-be drivers, many European cities slap high congestion fees on traffic in urban area to discourage driving, or ban private traffic from city centers entirely. (Vienna is an example of the latter trend.)
2) Suburban sprawl. The last fact, along with the plentiful availability of cheap land, means that US cities have tended to be built out rather than up. Home-ownership has been a much bigger historical trend in the US than in Europe (encouraged by heavy government subsidies for mortgages), and homes take up a lot of space.
The result has been sprawling suburban tracts built far from city centers. The distances involved make public transportation expensive to construct and relatively slow. And the sheer scope of development means that public transit can serve only a small fraction of the commuters who need to get to work (and the businesses and shops that spring up to cater to them). So it's simply not feasible to serve the entire population of most US cities with public transit. Cars are the only real option for most commuters.
Beside leading to a lack of coverage, sprawl kills the cost-effectiveness of public transit: because subway systems and the like have extremely high fixed costs, the only way to make them cost-effective is to serve a very large numbers of commuters. When the population of a city is geographically dispersed, a given rail stop will serve only a few commuters, and you'll never achieve the economies of density needed to pay for an extensive rail or subway system.
The BART system in the San Francisco Bay Area is a good example of the trade-offs that have to be made in response to US suburban sprawl: BART runs commuter trains that achieve very high speeds (you'll be zipping along faster than cars on the freeway beside you at some points) and traverse long distances. But it stops at only a few major population centers, runs infrequently, and is quite expensive, despite heavy tax subsidies. LA is even worse, because there's really no defined urban core there. The city is making a valiant attempt to promote its Metro system, but low population density and lack of common destinations makes rail lines very costly to expand on a per-capita basis.
3) Gasoline taxes. Gasoline taxes in the US are a fraction of those in most of Europe, leading to gas that's something on the order of half as expensive per gallon. The reasons for this are varied: the federal government has made no attempt to regulate carbon emissions through a carbon tax; other emission pollutants have been regulated through gasoline composition standards and catalytic converters rather than a Pigovian tax; and the gasoline tax here has never been intended to discourage driving in congested urban centers.
Since most of the cost of public transit is the fixed cost of constructing and maintaining rail lines and trains rather than the (almost negligible) marginal cost of running them, low gas prices reduce the relative cost-effectiveness of public transit. It's hard to convince a US commuter to pay $8+/day to ride a noisy, uncomfortable rail line into work, with connections on either end, rather than pay $5/day for gas to drive his comfortable, air-conditioned car. (And he has to own a car either way to get to the rail station and to run other errands, so he can't usually get rid of the fixed costs of car ownership by switching to public transit.) One can argue about whether driving a car is really a more comfortable way to travel than public transit, but most US commuters have been long conditioned with these preferences.
4) Costs of car ownership. Cars cost less (often much less) in the US, and yearly registration and maintenance fees are also lower. Parking is also much cheaper: houses come with parking, and most apartments outside urban centers offer parking for free.
I don't have a full explanation for the difference in car prices. Import tariffs may explain part of the difference, though whether these are driven by misguided trade policy or conscious congestion/environmental concerns, I don't know. Part of the explanation may also lie with the very robust market for used cars in the US, due to a steady supply from historical ownership patterns and well-developed systems for assessing quality (solving the "lemons problem") and connecting buyers and sellers. Whatever the reason, owning a car is simply much more affordable in the US than it is in Europe. (Financing options for purchasing or leasing cars are also prevalent in the US, though I'm less familiar with European options in this regard. But good financing options may make owning cars even more affordable for lower-income consumers in the US, conditional on the price of the car.)
It would take a more thorough historical analysis of car ownership patterns in the US and Europe than I've attempted to identify which of these factors is key to the underdevelopment of US public transit. But I've outlined a variety of reasons why we wouldn't expect transportation options to look the same in the two regions.
I'd also note that most of the rationales I've given don't apply to New York City: it's a very old city by US standards, is very densely developed, has relatively little access to nearby land for development, and has made aggressive attempts to reduce car congestion in Manhattan. Perhaps as a result, NYC has very well-developed public transit (though it still pales in comparison to, say, London). Further, many New Yorkers don't own cars, and some never learn to drive at all! NYC has experienced a more European history of urban development than the rest of the US, and so its transportation system looks more like Europe today.
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