This is the question of modern world history. In fact it is a huge set of questions on which a lot has been written. The Wikipedia article on Great Divergence gives a pretty good summary of some important work on the topic. I won't try to cover everything in there, but I will elaborate on a couple of key points that come to mind based on the original question.

First, we should keep in mind that Europe c. 1450 was not an especially advanced area at all. With the Dark Ages were coming to an end, Europe was perhaps a flourishing civilization, but not the only one. To its east, the Ottoman Empire was at the height of its power and expanding in Europe's direction. (The relevance of the Ottoman-European rivalry to our question has been getting more attention thanks to a recent book, How The West Came To Rule.) It was basically out of desperation to get around the Ottomans that Portuguese sailors began to make some breakthroughs in navigation. This gave them increased maritime contact with Africa, but despite their relatively advanced weaponry and navigation skills, the Portugese were not immediately able to dominate or conquer most of the peoples they found there.

Second, as the question already begins to address, Europe's ascent in the following centuries would not have been possible without the colonization of the Americas. Arguably the most important consequence of European conquest of the Americas was the massive influx of silver from South American mines under Spanish control. This was a central factor in the price revolution that shaped Europe's commercial development, and also in the course of relations between Europe and China. Among the other key reasons that European contact with the Americas mattered so much, aside from the silver, was the triangular trade that came to encompass the Atlantic.

This is the question of modern world history. In fact it is a huge set of questions on which a lot has been written. The Wikipedia article on Great Divergence gives a pretty good summary of some important work on the topic. I won't try to cover everything in there, but I will elaborate on a couple of key points that come to mind based on the original question.

First, we should keep in mind that Europe c. 1450 was not an especially advanced area at all. With the Dark Ages were coming to an end, Europe was perhaps a flourishing civilization, but not the only one. To its east, the Ottoman Empire was at the height of its power and expanding in Europe's direction. (The relevance of the Ottoman-European rivalry to our question has been getting more attention thanks to a recent book, How The West Came To Rule.) It was basically out of desperation to get around the Ottomans that Portuguese sailors began to make some breakthroughs in navigation. This gave them increased maritime contact with Africa, but despite their relatively advanced weaponry and navigation skills, the Portugese were not immediately able to dominate or conquer most of the peoples they found there.

Second, as the question already begins to address, Europe's ascent in the following centuries would not have been possible without the colonization of the Americas. Arguably the most important consequence of European conquest of the Americas was the massive influx of silver from South American mines under Spanish control. This was a central factor in the price revolution that shaped Europe's commercial development, and also in the course of relations between Europe and China. Among the other key reasons that European contact with the Americas mattered so much, aside from the silver, was the triangular trade that came to encompass the Atlantic.