SDI - serial digital interface
General SDI vs. HDMI Information

SDI is easier to connect to other professional equipment than HDMI. It is an all purpose signal, as opposed to HDMI that was originally intended only to connect to consumer display devices. There may be a slight advantage in quality where HDMI can be limited to 4:2:0 subsampling at certain resolutions, where SDI will usually be 4:2:2 or 4:4:4.

There isn't necessarily a performance advantage to SDI over HDMI. Sometimes HDMI outputs don't produce a clean HD signal and you need to put extra equipment in line to clean up the signal.

The disadvantage of SDI is cost. Most all SDI equipment is more expensive than HDMI equivalents. But you usually get higher quality equipment for that price.

In general, you need to weigh the exact specs of the equipment you are looking at for the application you want to use it for. If, in your case, the HDMI output has a 120fps maximum, but the SDI output has a 60fps maximum, you would want to choose the HDMI output.

One other often overlooked advantage of SDI, is that it uses locking BNC connectors (or sometimes other smaller form factor positive locking connectors) as opposed to non-locking connectors of HDMI. If you are going to be in an active environment, it offers a lot of piece of mind to know that your connector won't slip loose.

Why do you need an external output?

External outputs like HDMI or SDI are used for connecting a camera to peripheral devices. Some people use it to connect to an external monitor. External monitors can be bigger, allowing you to see more detail, or allow someone other than the camera operator to see the picture simultaneously.

Some people use it to connect to an external video recorder. External recorders can have larger storage capacity, have higher sustained I/O throughput, or capture a more advantageous format (less compressed, a different codec, etc.) than the internal SD card. For your application, it is the higher sustained throughput that might make a difference. The internal SD card, no matter which card class you get, probably will have a hard time recording high frame rate video flawlessly for extended periods of time. It might drop frames or just stop recording after buffers fill.

Decisions

You need to decide for what you are trying to do and what price/performance trade-offs you want to make. If you are doing work professionally, then I would strongly recommend going the SDI route. If you are just having fun with some short slow motion shots, then you probably only need the internal recorder. If you want to take a step up from there, still in the hobbyist realm, you could consider adding HDMI-based accessories in the future.

Lumix GH5 and the Panasonic YAGH Grip

According to this article, the YAGH Grip is not compatible with the GH5 and there is no planned SDI output.