In the context of US immigrant groups they’re almost the same, as there are huge amounts of Mexicans, Central Americans, Cubans, Dominicans, Puerto Ricans, and their descendants, but relatively few people from Spain, Portugal, or Brazil.
Argentina, for example, has a huge population of those from Italy. Brazil from Portugal, Mexico and central America from Spain. This is why you generally don't want to call an Argentine or Brazilian "Hispanic".
My guess as to why we call the continent "Latin America" is because for a long time Argentina dominated, being one of the richest and most advanced countries in the world, even giving birth to the expression "As rich as an Argentine". It is is still very important even today, but I don't want to get my Brazilian friends upset with me.
If you want to know why, in the US, we call someone an "X-American", whether that is a German-American, Latin-American, Hispanic-American, African-American, then you need to know where these terms come from. Words always have a historical, rather than a logical, explanation.
