The term "free software" was coined by Richard Stallman, and is associated with the Free Software Foundation. The term "open source" was coined by Eric Raymond and is advocated by him and other people at the Open Source Initiative. Nevertheless, those who consider themselves in either camp, much less those who use either or both terms, do not necessarily hold the opinions of these figures. Therefore I will not globally associate them with the "free software movement" or the "open source movement" because both include many users and developers with heterogeneous opinions on the subject. Moreover, they are pretty much one and the same.
Nevertheless, it is important to summarise their opinions, because they are recurring in many places.
Proprietary software is legal but illegitimate and immoral. Manufacturing and using proprietary software causes a lot of unhappy social and psychological side-effects. The knowledge that a software cannot be shared causes people to become reluctant to sharing, which is a natural and good part of living in a human society. The inability of people to modify software for their own needs makes them feel helpless, and at the mercy of external software.
Free software, on the other hand, is the natural conclusion derived from the basic facts of information, computing and software, and is highly moral. People, companies and other organizations can modify it, customise it and distribute it for their own use should the need arise, and so it actually benefits them.
Proprietary software is not illegitimate, just problematic from the economic sense. Open Source software gives many advantages to the end-users and is a generally a good thing. Copyleft licences are important in making sure certain software is not abused. It is not immoral to use proprietary software, it's just risky. Using or producing software that is not 100% open-source but pretty close, can be a good idea, depending on its licence and the general attitude of its developers.
This approach basically says this:
"I like free software because I can get a lot of useful software without charge. I may like contributing to free software because it helps other people, makes me happy and may indirectly benefit me technically or financially. But proprietary software is perfectly valid as well, if it's done right, and I may choose to use it or contribute to it.
In short: write code, use whatever tool you wish, and be happy."
The most prominent figure who holds this view is Linus Torvalds, but there are many others, some of them quite prominent. Such figures, however, tend to be less loud than the "religious" advocates of the other two views, and thus it may seem that they are at a minority. Part of the reason is that many of them inherently tend to value productive code and decisions over advocacy.
Note: I have prepared a longer (and unofficial) manifesto for this view which you may wish to consult for further information.
While some figures out there prominently stick to either ideology, most people hold a mixture of the three (or more?) approaches, or are just happy using free software or contributing to it, without thinking too much about its philosophy.
The terms themselves are used interchangeably by many people. "Open source" has become more common, partly because free software can mean software that is given free of charge. (the standard "free as in free speech" or "free as in free beer" distinction). Moreover, both the Free Software Foundation, and the people associated with the Open Source Institute are on friendly terms with each other and answer questions, give feedback, accept contributions, etc. from each other or from people that do not belong to either camp.
Like I said earlier, the fact that some licences would qualify as open-source and not as free software is usually a negligible fact. While some esoteric software has been released under custom licences that are open-source while not free software, most of the important software out there (and most software generally started by individuals) is free as well. [2]
[2] It is advisable not to use a custom licence anyhow, as this tends to confuse users and fellow developers. There are many common licences to choose from. Check the GNU licences list and the list of open source licences for such lists.