What is the difference between OctoPrint and OctoPi? Are they the same thing?

No, OctoPrint and OctoPi are not the same thing.

OctoPrint is a fully featured print server/web interface for 3D printers, written in Python and HTML/JS. It is platform agnostic and runs basically everywhere where you can install Python (the full version, not Micropython), be it a cheap single PCB computer like the Raspberry Pi running Linux, a fully featured gaming PC running Windows, a Mac running MacOSX or a server running some BSD flavor. It was created and is actively maintained by Gina Häußge aka @foosel.

OctoPi on the other hand is a preconfigured image for the Raspberry Pi. It is based on the Raspbian Lite image by the Raspberry Pi Foundation and bundles OctoPrint plus mjpg-streamer as a webcam server plus CuraEngine (removed in version 0.18) plus a couple of scripts to make your life easier and allow for easy initial setup. This image is probably the most common form of running OctoPrint. It was created and is actively maintained by Guy Sheffer aka @guysoft.

Think of it this way: OctoPi is more or less the operating system, OctoPrint is the main application that runs on it and actually talks to your printer.

Current versions:

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As I’m sure you know, Home Assistant has this same problem. Home Assistant was the actual software. Then they have HassOS which is the OS that runs on a Pi. Then they have Hassio which is Home Assistant and its addons running as Docker containers.

And the icing on top is that they literally just renamed Home Assistant into Home Assistant Core and Hassio into Home Assistant.

Super confusing.

Hopefully OP doesn’t need to go through anything like that.