undergraduate math vs graduate math

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.)

As a graduate student, the most useful skill I learnt as an undergraduate was not the mathematics itself, but how to learn mathematics. The edge of the subject is so wide that it's mostly not practical to get to a lot of current research problems as an undergraduate, even in a particular subfield like geometry or algebra for example.

That's not to say that the mathematics isn't important (and in fact I'm probably underplaying its importance because the parts of it I use all the time have become second nature), but knowing how to learn things efficiently is incredibly useful.

The graduate school experience probably varies quite a lot from university to university (and between countries as well), but my experience is similar to that described by Asaf in the comments - you still do some more formal courses at the start, but more independently than as an undergraduate, and at the same time your supervisor will suggest things you should read and problems you should think about - and these should lead to you discovering more things to read and problems to think about under your own volition.

I should probably also recount what I've always heard said by lecturers as the big difference between begin an undergrad and a grad student - as a grad student, you have to contribute original research. The upshot of this is that while the problems you see as an undergraduate may be difficult, they at least have answers, but this need not remain true when you are a grad student, and learning how to make judgements about which questions are worth persuing is an important aspect of postgraduate study - as mentioned by Eugene, Terence Tao has lots of good advice along these lines.

This also leads to a sound-bite answer to your question "why is graduate school even needed?" - because the process of learning how to do research is distinct from the process of learning how to do mathematics.

Terence Tao has great advice for mathematicians at every stage of their careers.

http://terrytao.wordpress.com/career-advice/