I am an incoming freshman and I wanted to know if there is any legitimate reason to take the honors math courses at Carnegie Mellon as opposed to the regular math courses?

TLDR; If you want a PhD in math, go all the way through. Else; at least do Matrix Theory.  Recall that the Honors Math sequence requires Matrix Theory (MT) and Vector Analysis (VA) in freshman year. Followed by two year-long courses in analysis and algebra (Math Studies). I'd say that you can partition the set of honors math people this way: people who take MT people who take both MT and VA people who do both MT and VA, but then sample Math Studies courses later on people who go through with the whole sequence. For those in 1, Matrix Theory yields a pretty huge benefit on its own. Most importantly, you get an solid foundation in linear algebra. This is essential because nearly all higher-level STEM courses require you to know linear algebra cold (and be able to prove stuff). Matrix Theory p-sets have fairly challenging problems that force you to write good proofs and understand the concepts thoroughly. Almost nothing in linear algebra takes that "long" to prove, so concision is emphasized.In 2, I would say that the additional workload of doing Vector Analysis without staying for Math Studies is a pretty low-return investment. Although the concepts in VA are pretty useful for physics/engineering/econ applications, most of them are covered again in Real Analysis 1/2 at a similar level.On the other hand, if you take some of the Math Studies sub-courses (e.g., Math Studies Algebra 1), then I'd say VA is worth it just because you get access to smaller classes with more rigorous material. Of course, the value of taking Math Studies is entirely dependent on one's ability to consume and enjoy said material. My understanding is that this isn't entirely obvious after finishing MT/VA.In 4, you have the real "mathletes," so to speak. If you're intent on going to grad school in math, this is the ideal subset to be in. You'll be well-trained in a wide range of topics at the graduate level. Many of our Math Studies graduates go on to top PhD programs. One thing to note is that if you want to study anything other than math in grad school, going all the way through Math Studies isn't the best idea. You'd be better off taking graduate courses in your department of interest.