I have copied and pasted Matt Emerton's excellent answer from the comments (1 2 3 4), which seems OK (with attribution) since this is community wiki anyway.
There are some areas of mathematics where the research is at a very high level, and where there are a fairly small number of leading experts who know each other more-or-less personally, so that ideas are shared not only through published literature but through more informal channels as well. In these circumstances, it can be hard to "break into" the area from the outside. The most basic reason is that in such situations one can find one's hard work being trumped by more general and powerful results that suddenly appear from the core group of researchers. Students who are working with the core group are hopefully spared this, b/c their advisor knows what everyone in the group is doing (more-or-less), and can try to protect their students a bit from being crushed by the other leaders in the field (through careful choice of problem, through communication with the leaders and letting them know what their students are working on, etc.). If your advisor is not connected to this core group (and it sounds like they are not) then they can't offer you this mantle of protection, and this sounds (to me) to be in part what your advisor is alluding too. I would ask your advisor to elaborate on/explain their comments (since it seems that you haven't fully understood whatever point they were trying to make). Also, if one of their concerns is the one I just described, this can be alleviated to some extent if you build your own personal connections with the area B core group. Just to conclude: math doesn't have to be a competition, but in old, highly developed areas, with a strong group of leaders who have a good grasp of where the field is moving and what results are and aren't currently in reach, it can be difficult for a newcomer to navigate and avoid having their work be literally trumped or at least overshadowed by the work of the core group. Since your advisor has explicitly said that area B is competitive, it seems that this is one of the concerns they have. So if you do move into area B, you should at least bear this in mind. One way to deal with this situation is to begin working on areas close to, but not exactly in, and not quite as mathematically central, as the key concerns of area B. This can provide you a chance to develop your technical skills, and then you can try to move into area B from a position of mathematical strength, with some solid experience already under your belt. I know several people who have done this successfully.