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.
Best Answer
I find I use conda first simply because it installs the binary, than try pip if the package isn't there. For instance psycopg2 is far easier to install in conda than pip.
https://jakevdp.github.io/blog/2016/08/25/conda-myths-and-misconceptions/
Pip, which stands for Pip Installs Packages, is Python's officially-sanctioned package manager, and is most commonly used to install packages published on the Python Package Index (PyPI). Both pip and PyPI are governed and supported by the Python Packaging Authority (PyPA).
In short, pip is a general-purpose manager for Python packages; conda is a language-agnostic cross-platform environment manager. For the user, the most salient distinction is probably this: pip installs python packages within any environment; conda installs any package within conda environments. If all you are doing is installing Python packages within an isolated environment, conda and pip+virtualenv are mostly interchangeable, modulo some difference in dependency handling and package availability. By isolated environment I mean a conda-env or virtualenv, in which you can install packages without modifying your system Python installation.
If we focus on just installation of Python packages, conda and pip serve different audiences and different purposes. If you want to, say, manage Python packages within an existing system Python installation, conda can't help you: by design, it can only install packages within conda environments. If you want to, say, work with the many Python packages which rely on external dependencies (NumPy, SciPy, and Matplotlib are common examples), while tracking those dependencies in a meaningful way, pip can't help you: by design, it manages Python packages and only Python packages.
Conda and pip are not competitors, but rather tools focused on different groups of users and patterns of use.