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

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 Avoid drawing conclusions from out-of-date information.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. PyPy doesn't support numpy. Some extensions are still not supported (Pandas, SciPy, etc.), take a look at the list of supported packages before making the change. Note that many packages marked unsupported on the list are now supported.
2. Python 3 support is experimental at the moment. has just reached stable! As of 20th June 2014, PyPy3 2.3.1 - Fulcrum is out!
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.

That site does not claim PyPy is 6.3 times faster than CPython. To quote:

The geometric average of all benchmarks is 0.16 or 6.3 times faster than CPython

This is a very different statement to the blanket statement you made, and when you understand the difference, you'll understand at least one set of reasons why you can't just say "use PyPy". It might sound like I'm nit-picking, but understanding why these two statements are totally different is vital.

To break that down:

• The statement they make only applies to the benchmarks they've used. It says absolutely nothing about your program (unless your program is exactly the same as one of their benchmarks).

• The statement is about an average of a group of benchmarks. There is no claim that running PyPy will give a 6.3 times improvement even for the programs they have tested.

• There is no claim that PyPy will even run all the programs that CPython runs at all, let alone faster.

Because pypy is not 100% compatible, takes 8 gigs of ram to compile, is a moving target, and highly experimental, where cpython is stable, the default target for module builders for 2 decades (including c extensions that don't work on pypy), and already widely deployed.

Pypy will likely never be the reference implementation, but it is a good tool to have.