Technically speaking there is very little difference between The Free Software Definition and The Open Source Definition (there are a few licenses which are Open Source but not Free Software). The difference lies in the underlying philosophy and reasons for their advocacy:

Free Software

This extract (1) from the FSF website helps to sum up the Free Software angle:
(...) ownership of a program--the power to restrict changing or copying it--is obstructive. Its negative effects are widespread and important. It follows that society shouldn't have owners for programs.

Another way to understand this is that what society needs is free software, and proprietary software is a poor substitute. Encouraging the substitute is not a rational way to get what we need.

RMS began the GNU Project and founded the FSF on his belief that Proprietary Software (and the power for authors of software to restrict its use) was immoral, and that all software should be free. Through advocating Free Software, he aims to make Proprietary Software obsolete.

Open Source

Open Source was a term created to help market Free Software to businesses. Companies are generally uninterested in the moral issues underlying the software they use. Open Source concentrates on the technical advantages that it is claimed the methodology brings: higher quality software, no vendor lock-in, lower cost, etc.

From the FAQ (2):

How is "open source" related to "free software"?

The Open Source Initiative is a marketing program for free software. It's a pitch for "free software" on solid pragmatic grounds rather than ideological tub-thumping. The winning substance has not changed, the losing attitude and symbolism have.

For Free Software advocates the choice is a moral one: Proprietary Software should not be used as it is immoral. For Open Source advocates, it is a technical one: Open Source leads to superior quality software.

Note that this is not to say that the advocates of Open Source do not agree with the principles of Free Software: they may simply choose the Open Source message as a more effective way to promote it. For example, Bruce Perens, who originally wrote the Open Source Definition, recently released an article entitled "It's time to talk about Free Software" (3) where he advocates discussing the Free Software viewpoint, now that there is enough interest in the subject.

Linus Torvalds is probably the embodiment of the Open Source viewpoint: he has publicly stated that he believes politics has no place within engineering. Although Linux is Free, he has demonstrated he has nothing against Proprietary Software, most notably through the use of the Bitkeeper System for holding the kernel source code.

RMS, however, believes this message is at complete odds with his own: he has stated that he would even use inferior software so long as it was free. With the massive growth of interest in "Open Source" over the past few years, it is little surprise that Free Software advocates have become even more vocal (some might even say extremist) as well.