Many of the answers here are out of date for 2015 (although the accepted one is not). Here’s the current state of things:

• Binary packages are now distributed as wheels (.whl files)—not just on PyPI, but in third-party repositories like Christoph Gohlke’s Extension Packages for Windows. pip can handle wheels; easy_install cannot.
• Virtual environments (which come built-in with 3.4, or can be added to 2.6+/3.1+ with virtualenv) have become a very important and prominent tool (and recommended in the official docs); they include pip out of the box, but don’t even work properly with easy_install.
• The distribute package that included easy_install is no longer maintained. Its improvements over setuptools got merged back into setuptools. Trying to install distribute will just install setuptools instead.
• easy_install itself is only quasi-maintained.
• All of the cases where pip used to be inferior to easy_install—installing from an unpacked source tree, from a DVCS repo, etc.—are long-gone; you can pip install ., pip install git+https://.
• pip comes with the official Python 2.7 and 3.4+ packages from python.org, and a pip bootstrap is included by default if you build from source.
• The various incomplete bits of documentation on installing, using, and building packages have been replaced by the Python Packaging User Guide. Python’s own documentation on Installing Python Modules now defers to this user guide, and explicitly calls out pip as “the preferred installer program”.
• Other new features have been added to pip over the years that will never be in easy_install. For example, pip makes it easy to clone your site-packages by building a requirements file and then installing it with a single command on each side. Or to convert your requirements file to a local repo to use for in-house development. And so on.

The only good reason that I know of to use easy_install in 2015 is the special case of using Apple’s pre-installed Python versions with OS X 10.5-10.8. Since 10.5, Apple has included easy_install, but as of 10.10 they still don’t include pip. With 10.9+, you should still just use get-pip.py, but for 10.5-10.8, this has some problems, so it’s easier to sudo easy_install pip. (In general, easy_install pip is a bad idea; it’s only for OS X 10.5-10.8 that you want to do this.) Also, 10.5-10.8 include readline in a way that easy_install knows how to kludge around but pip doesn’t, so you also want to sudo easy_install readline if you want to upgrade that.