How would the USA be different without the purchase of the Louisiana territory?

It is always difficult to answer counterfactual questions, because we can't know what would have happened. Maybe the best way to begin to answer this question is to point out a few things that the Louisiana Purchase did. First, it doubled the size of the United States, creating vast new swaths of land to be eventually settled and exploited for resources. Second, it opened the Mississippi River to navigation by American ships, which enabled the rapid settlement of the Northwest. This was probably the most significant effect of the Purchase. It eventually brought the United States into conflict with the various native peoples who inhabited the region. So what would have happened if the United States had not received the Louisiana Purchase? It is quite likely, especially given the importance of control of the Mississippi to the American economy, that the country would have eventually gained New Orleans, at least, by other diplomatic means, possibly including war. After the United States gained control of the Mississippi, there was really not much of an advantage to any European power in possessing the region between that region and the Rocky Mountains. So the Louisiana Purchase sped up a process that probably would have taken longer (but still would likely have happened) in any case. 
https://www.history.com/topics/westward-expansion/louisiana-purchase