Why shouldn't I use PyPy over CPython if PyPy is 6.3 times faster?

I've been hearing a lot about the PyPy project. They claim it is 6.3 times faster than the CPython interpreter on their site.

Whenever we talk about dynamic languages like Python, speed is one of the top issues. To solve this, they say PyPy is 6.3 times faster.

The second issue is parallelism, the infamous Global Interpreter Lock (GIL). For this, PyPy says it can give GIL-less Python.

If PyPy can solve these great challenges, what are its weaknesses that are preventing wider adoption? That is to say, what's preventing someone like me, a typical Python developer, from switching to PyPy right now?

NOTE: PyPy is more mature and better supported now than it was in 2013, when this question was asked. Avoid drawing conclusions from out-of-date information.

1. PyPy, as others have been quick to mention, has tenuous support for C extensions. It has support, but typically at slower-than-Python speeds and it's iffy at best. Hence a lot of modules simply require CPython. Check the list of supported packages, but look at the date that list was updated, because it's not not kept in lockstep with actual support, so it's still possible that packages that marked unsupported on that list are actually supported.
2. Python support typically lags a few versions behind, so if you absolutely need the latest features, you may need to wait a while before PyPy supports them.
3. PyPy sometimes isn't actually faster for "scripts", which a lot of people use Python for. These are the short-running programs that do something simple and small. Because PyPy is a JIT compiler its main advantages come from long run times and simple types (such as numbers). PyPy's pre-JIT speeds can be bad compared to CPython.
4. Inertia. Moving to PyPy often requires retooling, which for some people and organizations is simply too much work.

Those are the main reasons that affect me, I'd say.