What's the difference between "open with" and "recommended apps"? Some applications show up in the "Open With..." menu for a given type. But some applications that don't show up in that list still show up in the Recommended Apps list if you choose "Other..." and choose manually.

What is the difference between these two lists, and what would cause an app to appear in one but not the other?

(Lightroom is one example that doesn't appear in the Open With menu, but does show up in Recommended Apps.)
"Open with..." menu shows up the "by default" App that the extension of the file is associated with on the system, as well as the other Apps that the system is certain that can handle that file extension too. Example: An image .jpg. Can be by default the "Preview.app", but will also offer you (in the case you have those Apps) Photoshop, iPhoto, Internet browsers like Chrome, Firefox, Safari, etc. summarizing: any App that the system knows that can manage .jpg files.

The "Recommended Apps" list are the Apps of the "Open with..." menu + the rest of Apps that for some reason (mostly developers not specifying it) the system is not certain if is able to open it or not, then you can see some Apps "disabled" because the system knows that it cannot handle that extension. Example of a disabled App in this list is trying to open a .mp3 with the "Preview.app" which is intended to handle image file extensions and not audio file extensions.

Have in mind that the "Recommended Apps" list could offer Apps that are not able to handle the file you're trying to open because the system has no mean (for that App) to know if can or not.