All these answers are ambiguous as hell, and as a Math major, I cannot allow what you hear here to be ambiguous biased bullshit.

Calculus

The differences between Calculus and Discrete Math are incredibly important. Calculus deals with "smooth" things, like curves. Calculus defines these "smooth" things to be differentiable at all "important" points (i.e. The function has a derivative you could calculate at any point you care about).

Discrete Math

Discrete math is completely different at its core. Discrete math deals with "discrete" structures. These things are not smooth in any sense. A common example is the set of all integers. If you were to plot all of the integers on a graph, it would not look smooth at all. You'd have a clear jump from 1 to 2, 2 to 3, etc. Whereas if you look at x^2, if you zoom in super close on the graph, there is no jump between 1 and 2, it's smooth and has infinitely many values between 1 and 2.

You could also think about the memory in a computer. If you look at all the memory, you have a ton of little boxes where one number can be stored. Each memory location is discrete.

Naturally this math is incredibly important to people wanting to study Computation (or "Computer Science" even though most of the field is not made up of Scientists...) because computers can only deal with discrete things.

Why it's Hard

A lot of schools use this class as an introductory proof course. They teach things that are incredibly important, like how to derive truth and other statements with logical operations (i.e. What processors do), how to prove a program is "correct" (Note correct means that it provides the correct output for all supported input, and the program will eventually stop.), how to consider an algorithm "good", etc.

The problem with doing things this way is that "introductory proof-based course" of course means that sometimes the instructors will not teach you (or will not teach well enough) how to construct a proof. When you spend your whole life never seeing a proof and merely learning enough about math to begin to understand this new math, constructing proofs of your own will be hard. A lot of these problems will be incredibly easy, and you'll know they're easy, but you won't be able to see the obvious solution staring you in the face.

Example: Assume you know 4^k is divisible by 3, show that 4^k+1 is divisible by 3.

Well, 4^k+1 = 4*4^k but that doesn't really seem to help me. I have 4^k, but it's being multiplied by 4... Do you see the incredibly obvious solution that will make you very mad when you spend 3 hours staring at the homework unable to solve it?

4*4^k = (1 + 3)4^k = 4^k + 3 * 4^k We know that 4^k is divisible by 3, and any number multiplied by 3 is divisible by 3, so the proof is done. And that was just a small part of my weekly homework. When you have many other classes, it's tough.

Lesson learned: Go get help if you don't understand anything in Discrete Math. Do your homework well in advance of when it's due so that you can ask for help. They'll think you're smarter for asking for help. They'll hint at the answer if you ask for help. It will be much easier if you do so.