Political scientists and legal scholars like to refer to this as "laboratories of democracy." While the federal government has a lot of power and gets all the press, each state really makes the vast majority of its own decisions. This allows each state to try something new, without foisting the idea on the entire nation. So, if it works, then other states may adopt it, and if not, it goes away. In theory, this allows for legal and policy innovation on a wide scale.
For example, one of the health care proposals currently airing in Washington DC is to require all Americans to buy health insurance. This ides comes directly from one done in Massachusetts, and the proponents of this plan trot out the "success" in Mass. to promote the idea.
With respect to competition for business and jobs, this certainly happens. Many southern states (TN, MS, AL, LA) have vehemently competed to try and get new car manufacturing plants in their localities. Indeed, counties and cities within each state have competed too. These states/counties/cities have offered tax incentives, but the state laws also are much more anti-union (or less pro-union, if you prefer) than states such as Michigan or Ohio. So, not only are these states "competing" through tax policy, but they are also doing so via law.
There are tradeoffs to be sure--Michigan might lose some jobs to Tennessee, but the Michiganders like their unions and don't want to offer tax subsidies and are thus willing to make that trade. Conversely, Tennesseeans might not have the job security and bargaining power that comes with a union shop, and are subsidizing the new plant via lost tax revenue, but they have made the decision to forgo those benefits to obtain more jobs.
Also, the mobility of American labor is high, but often depends on class. People move for new jobs all the time, but there are structural considerations--the cost of moving/selling your house, the social cost of losing your social network at your home, etc. Statistically speaking, richer and more educated are more likely to move for a job, because they are able to do so. But poorer people are less likely to have the resources for such a move, let alone the job skills to obtain a job across the country. This is in contrast to the 1950s and 60s when huge numbers of poorer--and largely African-American--people moved from the South to Northern cities for industrial jobs, when those Northern cities needed a larger labor pool.
Perhaps I'm way off topic. But maybe this sheds some light on Orathaic's initial question.