The Valve Index is just an I/O device (display, tracking* and controllers), whereas the Steam Frame is also a stand-alone Linux computer (although with a very limited battery life and around the power of a cellphone).

For tracking, the Index does room-scale outside-in tracking using little boxes that you have to hang up in your room, called Lighthouse VR base stations. The Frame uses inside-out tracking of controllers using a combination of IR cameras and gyro--maybe not quite as accurate, but significantly more convenient. I believe that the original base stations that were sold with the index are no longer being made, but other ones are, though I'm not sure.

The Frame supports eye-tracking, plus eye-tracking features like foveated rendering and streaming. The Index doesn't, at least not out of the box. These optimizations essentially make things look better/sharper exactly where you're currently looking even as you look around, which generally gives you better performance.

The Frame can be used wirelessly, and comes with a dedicated 6GHz streaming dongle. The Frame also supports an interesting mode called a Multi-Link connection which uses both the dongle and your WiFi at the same time to (in theory) give you a most stable connection. Unlike the Frame, the Index must be plugged into a computer at all times.

The Frame is lighter and smaller than the Index, and has better, higher-resolution screens.