Difference between Pure and Applied Mathematics?

So, I have been looking into college courses to see what I want my major to be, and I noticed that M.I.T offers two specified types of Mathematics. Pure and Applied. What is the major differences in Pure vs. Applied and what are examples of each?

The lines between these two divisions, much as in all divisions within mathematics, is often blurred. However, the traditional division between the two is that Applied Mathematics has a very clear connection to physical real-world problems. At its heart are PDE's, but also included are things like numerical methods and (once upon a time) what are now called computer science and statistics.

Pure Mathematics is mathematics for its own sake, pursuing questions based on the internal attractiveness of the questions. At its heart is number theory.

The best way to tell the difference between "pure" and "applied" mathematics is to compare the kinds of problems faced in each. Consider the following list of questions:

1. How can we model, mathematically, the movements of a human heart?
2. How quickly might a particular virus spread through a population of people?
3. What's the best way to throw a skipping stone? That is, how can we maximize the number of bounces on the surface of the water before the stone sinks?
4. Why do whirlpools form? Is there an optimal strategy to escape from one?
5. How can we accurately predict what the weather will be like in 3 days time? Is it possible to accurately predict what the weather will be like in 3 months time?
6. Why do soap bubbles form spheres? In cases where they do not form spheres (YouTube link), what kinds of shapes might they form?
7. Consider $k$ runners on a circular track of unit length. At time $t = 0,$ all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely at time $t$ if the distance along the track from them to the nearest other runner is at least $1/k$ at time $t.$ For which values of $k$ is every runner lonely at some time?
8. For which $m\times n$ chessboards does there exist a knight's tour? What about if the knight is allowed to go off one side of the chessboard and come back on the other side (cf. Asteroids)?
9. What is the area of a circle of radius $r,$ and what's its perimeter? Can we find a relationship between the two that also relates the volume of a sphere of radius $r$ to its surface area?
10. Can we find integers $a$ and $b$ with $b\neq0$ such that $\pi=a/b?$ What about if we replace $\pi$ by $\sqrt{2}?$
11. Given a number $n,$ is there some easy way to compute the number of prime numbers less than $n?$ Is there some easy way to estimate the greatest prime less than $n?$
12. Pick a positive integer. If it's even, then divide it by $2;$ if it's odd, then multiply it by $3$ and then add $1.$ For which starting values do we eventually reach $1$ as a result of repeating this process?

All of these questions have some kind of mathematical answer. I would say the first half are "applied" problems and the second half are "pure" problems. Broadly speaking, a problem is "applied" if it is phrased in terms of physical phenomena. That being said, question 6 leads to some very interesting pure mathematics, and question 7 has been clearly stated in terms of physical, everyday things. Hopefully, this is enough indication that there is not always such a clear distinction between "pure" and "applied."

It should be noted that I have certainly failed to give an indication of the breadth of either area: applied mathematics is much more than those six questions above, and pure mathematics is much more than the other four. When the time comes, I'm sure you'll be able to find more examples of the kinds of problems solved in mathematics by looking through your university library. I don't see any reason why you should choose in advance; I advise you to keep your options open, if you can.

EDIT: I have just realized that I am more than a year late to the party. Nevertheless, my answer might be helpful to someone, so I'll leave it here.

It can be boiled down to (my choice of) Algebra, Number Theory, Topology and Gemoetry versus Statistics, Mechanics and Applications ot Physics.

I would have to say that pure mathematics involves pure numbers (and other objects that don't have units of measurement) while applied mathematics involves quantities (numeral values and units of measurement such as volts or dollars).

For example, if you are studying physics or statistics without using any units of measurement, then these would be forms of pure mathematics (mathematical physics and mathematical statistics). But if you are studying them using units of measurement, then they are applied mathematics (applied physics and applied statistics).

As previous answers mentioned, there is a lot of overlap between Pure Mathematics and Applied Mathematics because they might use similar techniques or concepts. What makes them different is the goals set on in Applied Mathematics versus Pure Mathematics, I am just going to refer them to AM and PM respectively. For AM the goal is typically to advance mathematics for the sake of some practical purpose, now this might still be quite theoretical, for instance people in mathematical physics might be creating new math for the sake of advancing our understanding the physical world. While in PM is advanced for the sake of advancing mathematics without a concern for its practical applications, even if practical applications do exist or not.

What is important to also emphasize is that one field is not necessarily more difficult than the other one. For instance there are mathematicians who study the mathematical ideas behind General Relativity, this involves very technical concepts that require studying a lot of PM but serves AM as it allows us to have better physical understanding of our universe. Whether PM is harder or not than AM I believe is not a very useful conversation though.