Yes, they exist.
When we call software “free,” we mean that it respects the users' essential freedoms: the freedom to run it, to study and change it, and to redistribute copies with or without changes. This is a matter of freedom, not price, so think of “free speech,” not “free beer.”
Since the obvious meaning for “open source” is not the meaning that its advocates intend, the result is that most people misunderstand the term. According to writer Neal Stephenson, “Linux is ‘open source’ software meaning, simply, that anyone can get copies of its source code files.” I don't think he deliberately sought to reject or dispute the official definition. I think he simply applied the conventions of the English language to come up with a meaning for the term. The state of Kansas published a similar definition: “Make use of open-source software (OSS). OSS is software for which the source code is freely and publicly available, though the specific licensing agreements vary as to what one is allowed to do with that code.”
Source: Why Open Source misses the point of Free Software
open source was kind of created because
free software is not a very good term to use in a business world (source). Open source is a bit less restrictive than free software. Because every piece of free software can be classified as open source software too (taken from the content of their definitions here and here), it's perfectly normal that the term open source is more widely used.
Free software's essential freedoms
A program is free software if the program's users have the four essential freedoms:
- The freedom to run the program as you wish, for any purpose (freedom 0).
- The freedom to study how the program works, and change it so it does your computing as 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.
Open source criteria
- Free Redistribution
- Source Code
- Derived Works
- Integrity of The Author's Source Code
- No Discrimination Against Persons or Groups
- No Discrimination Against Fields of Endeavor
- Distribution of License
- License Must Not Be Specific to a Product
- License Must Not Restrict Other Software
- License Must Be Technology-Neutral
Free software's freedom 0 requires OSD criteria 5, 6, 7, 8, 9, and 10; freedom 1 requires OSD criterion 2 (and possibly 7); freedom 2 probably requires OSD criteria 1, 7, 8, and 9; and freedom 3 probably requires OSD criteria 1, 2, 3, 4, 7, 8, and 10.
Therefore, every free software is open source software and satisfies all of its criteria. The reverse doesn't have to be the case.
As an example, NASA's Open Source Agreement (NOSA) is approved by the OSI (source) but not approved by the FSF because it allows only contributions that are developed by the contributor and doesn't allow contributions that use the code developed by some third party (source). Therefore, every piece of software that uses NOSA as its license is open source, but not free software.
To sum it up, open source means that the users have the access to the source code, but free software means that the users are free to use the code in any way they want. If something is considered as a free software, you can also use the term open source for it. But not every piece of open source software is actually free software. There's that big difference between the terms that confuses people.