(Paper submission) Institution vs affiliation
Institution usually refers to the institution that the work was done at, or at least who pays the bills for that particular author most of the time.  For many authors this is the only institution they are affiliated with, so it really makes no difference.
Affiliation refers to other organizations that the author is affiliated with.  For instance, say I am a researcher at university A, but a medical doctor at an associated hospital (I'm not, but just say...).  Then I might list the university as my institution (because they paid for my lab space, perhaps they had the research ethics board etc) but I am affiliated with the hospital as well, which is where the research subjects came from.  I would list both, because both organizations should be recognized for their financial or in-kind contributions to the work.
Sometimes, authors have multiple affiliations just because they've been given appointments by other organizations, whether because they want to boost their prestige or because there was a relevant reason for this to happen.  For instance, if a university researcher is working on a space mission and they are given appointments by some NASA research centers, they might list those appointments as their affiliations.  These organizations may have contributed lab space, equipment, funding, personnel etc, all under the guidance of the researcher, and the organizations should therefore be listed.
Sometimes it is necessary to divulge a conflict of interest as well.  Say that I am a university researcher doing research on a drug for a company that I am also on the board of directors for.  I should mention that company in my affiliations (among other things)