That was me. I grew up believing I hated math. Struggled all the way through middle & high school to AP calc and just found it incredibly boring and tedious. Ended up opting out of doing engineering/science in undergrad because I just couldn't stand doing all the math.
Long story short, years later ended up going back to school for CS and took discrete math as one of my first courses, and remember being blown away by how cool it was. All this time thinking I hated math!
Hard to say exactly what the difference is. Partially I think my brain just groks discrete concepts more easily.
But also the class had a heavy emphasis on proofs, which I think was really important. At a certain level this type of problem-solving can start to resemble philosophy. Chugging through a proof, figuring out just the right way to construct it and slapping a triumphant "Q.E.D." at the end is an empowering experience, especially the first time. There's a world of difference between "you throw a ball, solve for its velocity at time x" and "prove that there must be a ball" (I'm embellishing of course). It's a difference between obtaining an answer for a specific instance of a situation, and shedding light on some fundamental/universal property of the world. To me that feels profound in a sense, which makes it exciting.
Proofs don't belong solely to the domain of discrete math, of course, so this probably isn't as much a testament to the subject as it is to the general problem-solving approach. It would be nice if students could get exposed to this a bit earlier, I think there are many folks like myself who would realize that they can love math too.