What are the continuities and changes in Latin America's political and economic systems between 1800 and the present?
This is a huge question because it covers many countries and more than two centuries. Therefore, any answer can only be a very gross simplification of the actual truth.
There have been two major changes in the political systems of most Latin countries. First, they have all become independent. Second, many of them have become democracies at least in name. The major continuity is that many of them have remained less than truly democratic. Until recently, Mexico was a one-party state. Venezuela is ostensibly a democracy but Chavez is a strong man. These are some examples of the lack of true democracy in the region.
Economically, the major change is that the Latin countries have become richer than they had been. The major continuity is that many of them are still in what some would call a neo-colonial relationship with the US and other rich nations. Many Latin countries still have economies based on selling raw materials to more developed countries.