OpenWRT and DD-WRT for example supply two files:
- factory (.img)
- sysupgrade (.bin)
Why is one a .img and the other a .bin?
Does it matter which one I just and if yes, when should I choose which?
OpenWRT and DD-WRT are just two different projects that "power" routers. Look at it as if you're comparing Linux distributions. Your question would be similar to asking what the difference between Ubuntu and Fedora is. They offer different features for different purposes.
About the files: One is an .img file, which completely erases the firmware and replaces it with a new image. The .bin is just a "sysupgrade" so it's just an upgrade that doesn't replace the current image.