Open Source but not Free Software (or vice versa)
- by TRiG
The definition of "Free Software" from the Free Software Foundation:
  “Free software” is a matter of
  liberty, not price. To understand the
  concept, you should think of “free” as
  in “free speech,” not as in “free
  beer.”
  
  Free software is a matter of the
  users' freedom to run, copy,
  distribute, study, change and improve
  the software. More precisely, it means
  that the program's users have the four
  essential freedoms:
  
  
  The freedom to run the program, for
  any purpose (freedom 0).
  The freedom
  to study how the program works, and
  change it to make it do what you wish
  (freedom 1). Access to the source
  code is a precondition for this.
  The
  freedom to redistribute copies so you
  can help your neighbor (freedom 2).
  The freedom to distribute copies of
  your modified versions to others
  (freedom 3). By doing this you can
  give the whole community a chance to
  benefit from your changes. Access to
  the source code is a precondition for
  this.
  
  
  A program is free software if users
  have all of these freedoms. Thus, you
  should be free to redistribute copies,
  either with or without modifications,
  either gratis or charging a fee for
  distribution, to anyone anywhere.
  Being free to do these things means
  (among other things) that you do not
  have to ask or pay for permission to
  do so.
The definition of "Open Source Software" from the Open Source Initiative:
  Open source doesn't just mean access
  to the source code. The distribution
  terms of open-source software must
  comply with the following criteria:
  
  
  Free Redistribution The license shall not restrict any party from
  selling or giving away the software as
  a component of an aggregate software
  distribution containing programs from
  several different sources. The license
  shall not require a royalty or other
  fee for such sale.
  Source Code The program must include source code, and must allow
  distribution in source code as well as
  compiled form. Where some form of a
  product is not distributed with source
  code, there must be a well-publicized
  means of obtaining the source code for
  no more than a reasonable reproduction
  cost preferably, downloading via the
  Internet without charge. The source
  code must be the preferred form in
  which a programmer would modify the
  program. Deliberately obfuscated
  source code is not allowed.
  Intermediate forms such as the output
  of a preprocessor or translator are
  not allowed.
  Derived Works The license must allow modifications and derived works,
  and must allow them to be distributed
  under the same terms as the license of
  the original software.
  Integrity of The Author's Source Code The license may restrict
  source-code from being distributed in
  modified form only if the license
  allows the distribution of "patch
  files" with the source code for the
  purpose of modifying the program at
  build time. The license must
  explicitly permit distribution of
  software built from modified source
  code. The license may require derived
  works to carry a different name or
  version number from the original
  software.
  No Discrimination Against Persons or Groups The license must not
  discriminate against any person or
  group of persons.
  No Discrimination Against Fields of Endeavor The license must not restrict
  anyone from making use of the program
  in a specific field of endeavor. For
  example, it may not restrict the
  program from being used in a business,
  or from being used for genetic
  research.
  Distribution of License The rights attached to the program must apply to
  all to whom the program is
  redistributed without the need for
  execution of an additional license by
  those parties.
  License Must Not Be Specific to a Product The rights attached to the
  program must not depend on the
  program's being part of a particular
  software distribution. If the program
  is extracted from that distribution
  and used or distributed within the
  terms of the program's license, all
  parties to whom the program is
  redistributed should have the same
  rights as those that are granted in
  conjunction with the original software
  distribution.
  License Must Not Restrict Other Software The license must not place
  restrictions on other software that is
  distributed along with the licensed
  software. For example, the license
  must not insist that all other
  programs distributed on the same
  medium must be open-source software.
  License Must Be Technology-Neutral No provision of the license may be
  predicated on any individual
  technology or style of interface.
  
These definitions, although they derive from very different ideologies, are broadly compatible, and most Free Software is also Open Source Software and vice versa. I believe, however, that it is possible for this not to be the case: It is possible for software to be Open Source without being Free, or to be Free without being Open Source.
Questions
Is my belief correct? Is it possible for software to fall into one camp and not the other?
Does any such software actually exist? Please give examples.
Clarification
I've already accepted an answer now, but I seem to have confused a lot of people, so perhaps a clarification is in order. I was not asking about the difference between copyleft (or "viral", though I don't like that term) and non-copyleft ("permissive") licenses. Nor was I asking about your personal idiosyncratic definitions of "Free" and "Open". I was asking about "Free Software as defined by the FSF" and "Open Source Software as defined by the OSI". Are the two always the same? Is it possible to be one without being the other?
And the answer, it seems, is that it's impossible to be Free without being Open, but possible to be Open without being Free. Thank you everyone who actually answered the question.