The standard reply you'll get when you ask people is that since you had to return to get back to earth, at that point you had to accelerate, and thus special relativity isn't valid anymore, and the problem has to be solved with general relativity.
This, however, is not correct. The truth is that the equation for time dilation you are using is only valid for the above, simplified case of constant movement with no acceleration. So here's the short reply: When the twin is returning home, the inertial system you've been using CHANGES. It suddenly moves in a different direction - that's not something inertial systems generally do ;), and is the cause of the problem: In your calculation, you've been assuming that you're calculating stuff in a inertial system, but you're not. That false assumption leads to the wrong result.


EDIT:



Nebolous statements like that really don't help much, don't ya think?
Relativity predicts TWO effects that change the "flow of time", if you will. One is described by special relativity - what is usually referred to as time dilation. It can be summarized as "Moving clocks go slower".

The second effect is due to general relativity. Masses curve spacetime, in other words, it changes the geometry of it. The time you're measuring (the "proper time") is nothing but the length of your path through spacetime (disregarding units). Now, obviously, if masses curve spacetime, it only makes sense that the lengths of your paths change, too (and thus your measured proper time).

Whats fascinating is that for satellites, bothe effects actually work in different ways: They're moving quiet fast relative to the earth, so that makes their clocks go SLOWER, yet at the same time, the mass of earth makes their clocks go FASTER compared to the one of someone on earth (the closer you are to masses, the slower your clocks go).