Quoting from the Conda blog:
Having been involved in the python world for so long, we are all aware of pip, easy_install, and virtualenv, but these tools did not meet all of our specific requirements. The main problem is that they are focused around Python, neglecting non-Python library dependencies, such as HDF5, MKL, LLVM, etc., which do not have a setup.py in their source code and also do not install files into Python’s site-packages directory.
So Conda is a packaging tool and installer that aims to do more than what pip does; handle library dependencies outside of the Python packages as well as the Python packages themselves. Conda also creates a virtual environment, like virtualenv does.
As such, Conda should be compared to Buildout perhaps, another tool that lets you handle both Python and non-Python installation tasks.
Because Conda introduces a new packaging format, you cannot use pip and Conda interchangeably; pip cannot install the Conda package format. You can use the two tools side by side (by installing pip with conda install pip) but they do not interoperate either.
Since writing this answer, Anaconda has published a new page on Understanding Conda and Pip, which echoes this as well:
This highlights a key difference between conda and pip. Pip installs Python packages whereas conda installs packages which may contain software written in any language. For example, before using pip, a Python interpreter must be installed via a system package manager or by downloading and running an installer. Conda on the other hand can install Python packages as well as the Python interpreter directly.
and further on
Occasionally a package is needed which is not available as a conda package but is available on PyPI and can be installed with pip. In these cases, it makes sense to try to use both conda and pip.