Both StaticPython and PyRun are free, portable binary distributions of Python 2.x and 3.x, for Linux, Mac OS X (and PyRun also works on FreeBSD without Linux emulation). That is, a user without root access can quickly download, extract and run either StaticPython or PyRun, and it won't conflict with the (possibly old) Python versions installed to the system (by root). Windows users should use Portable Python instead.
An important difference is that StaticPython doesn't need any external files to run on Linux, while PyRun needs lots of system libraries (libssl.so.1.0.0, libcrypto.so.1.0.0, libz.so.1, libsqlite3.so.0, libz2.so.1, libpthread.so.0, libdl.so.2, libutil.so.1, libm.so.6, libc.so.6), and will not work on systems which don't have all of the required libraries installed (with the required version), so PyRun is much less portable.
Another important difference is that with PyRun you can compile and install C extensions, and with StaticPython you can't (even ctypes doesn't work with StaticPython). However, many extensions are precompiled: Stackless + greenlet + libssl + Tokyo Cabinet + libevent2 + Concurrence + AES + Syncless + MessagePack + Gevent MySQL + PyCrypto.
Another important difference is that PyRun integrates nicely with packaging (setuptools, pip etc.), and is a nice, self-contained replacement for virtualenv. StaticPython doesn't help you with packaging, you have to do Python library installation and distribution manually.
There are many small differences, e.g. PyRun is available for both 32-bit and 64-bit, ans StaticPython is 32-bit only. PyRun needs at least 14 megabytes of disk space (10 megabytes for the binary executable and about 4 megabytes for the mandatory system library dependencies), and StaticPython needs between 6 and 9 megabytes depending on which feature set you need.
My recommendation: If PyRun works on your system (because you have the right version of system libraries installed, and you have enough free disk space), then use PyRun, otherwise use StaticPython.
Caveat: I am the author of StaticPython.