Here is what I tell my grad students:
The difference between undergrad mathematics and graduate mathematics is the difference between art history, or art appreciation, and learning to be an artist.
As an undergraduate you see a lot of mathematics, but you don't create new mathematics. The goal of graduate school (and here I am speaking from experience with top fifty U.S. graduate schools, so what I am saying probably applies best in that context) is to learn how to create new mathematics, and then to create that new mathematics.
One specific consequence of this (in my view) is the following: often in undergraduate mathematics classes, proofs and rigor are presented almost as moral imperatives --- as if it is a moral failing to know a statement without knowing why it is true; consequently, people often put a lot of effort into learning arguments just for the sake of having learnt them. (This is exaggerated, perhaps, but I think it reflects something real.) On the other hand, in research, one learns arguments for different reasons: to learn technique, to pick out important ideas --- there is a professional aspect to the way one looks at pieces of mathematics which is not usually present in undergraduate mathematics. One gives proofs in order to be sure that one hasn't blundered; one's interaction with the mathematics and the arguments is much more visceral than in undergraduate courses.
(I am not speaking from any experience now, but I think of the difference between learning how to interact with a block of marble, and bring a new form out of it, however rough it might be, in comparison to looking and learning about a lot of existing beautiful statues, masterpieces that they are.)
Best Answer
I strongly urge you to read an answer that I wrote up for another question: (link).
The landscape of higher education is changing rapidly right now, and the path to becoming a tenured professor is not very similar anymore to what it was even just 15 years ago. I believe (others will disagree) that this process will change even more dramatically in the next 15 years.
If math is what you love, you should definitely continue studying it and pursue it to see what options are available to you. But don't disregard the need to assess your aptitudes and honestly appraise your skill next to the skill of other PhD quality mathematicians vying for the same faculty jobs.
Along these lines, I personally think it's better to go for an applied course of study. Devote time to learning excellent programming skills in multiple languages, as well as non-trivial software design skills. Knowing how to tinker in Matlab, Maple, and Mathematica is not worth anything. Similarly, learn advanced statistics. Study what people do with large data sets (mostly computational Bayesian methods these days). Learn about scientific computing and implementational details.
Additionally, choose a topical domain for which you believe the job outlook will be good. This could be computational finance, computational biology, applied machine learning, or a host of others about which I am less familiar.
Ask your current professors for advice on this. But be careful. People who were lucky enough to make it to the position of professor often suffer from narrative fallacy and selection bias. That is, rather than acknowledging that they are not much more skillful than peers who were not able to win faculty jobs, professors tend to attribute their fortunate position to various narrative stories about what they specifically did to work hard and achieve things. But what worked for one person in one situation is too idiosyncratic for you to care about; it doesn't describe what works for general situations, nor for the future situation that will be relevant to you.
Do lots of research before committing yourself to one direction or another. And consider many other important life factors, not just how much you like math, such as:
These things matter a great deal in your decision in college to orient yourself towards a future career. Most people will give advice to you in far-thinking mode, but this isn't a good thing. You should realize that the economic landscape of the world in 5-15 years will determine what jobs you have the option of doing. That's just an attribute of reality. And the more time you spend reflecting on that and planning for what reality will be like, the better suited you'll be to try to make your own goals happen. And, more importantly, the better capable you'll be of re-orienting your goals to match with what is possible.