Let's say your fellow zoologists and you had decided to spend a few weeks in an African national park to look very carefully, with notepads and cameras in hand, at prides of lions, analysing their hunting patterns and discovering their strategies for hunting prey. You were "observing" the lions by watching them in a purposeful manner in order to learn something, conduct a scientific study or obtain some useful information.
Let's say your friends and you were tourists on a safari tour in South Africa, and you were looking at some lions which happen to be occupying a stretch of a road. The jeep pulled up next to these majestic beasts and you took pictures while looking at two cubs playfully wrestling each other. You were "watching" the cubs which had captured your attention and you were keen to see what would happen next.
In your sentence: "We spent weeks in Africa________________________ the way lions catch their prey.", I would personally choose "observing" as "the way" suggests that you were trying to learn something specific by looking at the lions. I cannot say that "watching" is wrong, but based on the contextual cue, "observing" would be more context-appropriate.
I hope this helps.