Besides the COW difference, STL containers are much more widely supported on a variety of platforms. Qt is portable enough if you limit your work to "mainstream" platforms, but the STL is available on many other more obscure platforms too (e.g., Texas Instruments' DSPs).
Because the STL is standard rather than controlled by a single corporation, there are, generally speaking, more programmers who can easily read, understand, and modify STL code and more resources (books, online forums, conferences, etc.) to support them in doing this than there are for Qt. That's not to say that one should shy away from Qt for this reason alone; just that, all other things being equal, you should default to the STL, but of course all things are rarely equal, so you'll have to decide in your own context which makes the most sense.
In regard to AlexKR's answer: the STL performance is guaranteed within limits, but a given implementation may make use of platform-dependent details to speed up their STL. So in that sense, you may get different results on different platforms, but it will never be slower than the explicit guarantee (modulo bugs).