Check out this article. It gave me a lot more information on the difference between these two motivation theories and specifically how they can be applied in the classroom.
The article states that motivation is not interest but rather interest is an influence on motivation. A child may have an interest in cars, but that does not necessarily mean that they are intrinsically or extrinsically motivated. Intrinsic motivation is situational, and therefore can change over time. The article states that "doing something because one wants to can easily become doing it because one needs to".
Overall, children who are intrinsically motivated engage in activities that enhance learning, which leads to greater self-efficacy and less anxiety. These are the children who love learning and become life-long seekers of knowledge. These are also the children who are capable of setting goals for themselves, and working, through intrinsic motivation, to achieve their goals. As teachers, if we can provide engaging, applicable learning experiences for kids, I think that kids will be able to develop more of a vision for their life early on, and will hopefully be given the tools to reach their goals as well.